Special¶
DiracDelta¶
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class
sympy.functions.special.delta_functions.DiracDelta(**kwargs)[source]¶ The DiracDelta function and its derivatives.
Explanation
DiracDelta is not an ordinary function. It can be rigorously defined either as a distribution or as a measure.
DiracDelta only makes sense in definite integrals, and in particular, integrals of the form
Integral(f(x)*DiracDelta(x - x0), (x, a, b)), where it equalsf(x0)ifa <= x0 <= band0otherwise. Formally, DiracDelta acts in some ways like a function that is0everywhere except at0, but in many ways it also does not. It can often be useful to treat DiracDelta in formal ways, building up and manipulating expressions with delta functions (which may eventually be integrated), but care must be taken to not treat it as a real function. SymPy’soois similar. It only truly makes sense formally in certain contexts (such as integration limits), but SymPy allows its use everywhere, and it tries to be consistent with operations on it (like1/oo), but it is easy to get into trouble and get wrong results ifoois treated too much like a number. Similarly, if DiracDelta is treated too much like a function, it is easy to get wrong or nonsensical results.DiracDelta function has the following properties:
\(\frac{d}{d x} \theta(x) = \delta(x)\)
\(\int_{-\infty}^\infty \delta(x - a)f(x)\, dx = f(a)\) and \(\int_{a- \epsilon}^{a+\epsilon} \delta(x - a)f(x)\, dx = f(a)\)
\(\delta(x) = 0\) for all \(x \neq 0\)
\(\delta(g(x)) = \sum_i \frac{\delta(x - x_i)}{\|g'(x_i)\|}\) where \(x_i\) are the roots of \(g\)
\(\delta(-x) = \delta(x)\)
Derivatives of
k-th order of DiracDelta have the following properties:\(\delta(x, k) = 0\) for all \(x \neq 0\)
\(\delta(-x, k) = -\delta(x, k)\) for odd \(k\)
\(\delta(-x, k) = \delta(x, k)\) for even \(k\)
Examples
>>> from sympy import DiracDelta, diff, pi >>> from sympy.abc import x, y
>>> DiracDelta(x) DiracDelta(x) >>> DiracDelta(1) 0 >>> DiracDelta(-1) 0 >>> DiracDelta(pi) 0 >>> DiracDelta(x - 4).subs(x, 4) DiracDelta(0) >>> diff(DiracDelta(x)) DiracDelta(x, 1) >>> diff(DiracDelta(x - 1),x,2) DiracDelta(x - 1, 2) >>> diff(DiracDelta(x**2 - 1),x,2) 2*(2*x**2*DiracDelta(x**2 - 1, 2) + DiracDelta(x**2 - 1, 1)) >>> DiracDelta(3*x).is_simple(x) True >>> DiracDelta(x**2).is_simple(x) False >>> DiracDelta((x**2 - 1)*y).expand(diracdelta=True, wrt=x) DiracDelta(x - 1)/(2*Abs(y)) + DiracDelta(x + 1)/(2*Abs(y))
See also
Heaviside,sympy.simplify.simplify.simplify,is_simple,sympy.functions.special.tensor_functions.KroneckerDeltaReferences
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classmethod
eval(arg, k=0)[source]¶ Returns a simplified form or a value of DiracDelta depending on the argument passed by the DiracDelta object.
- Parameters
k : integer
order of derivative
arg : argument passed to DiracDelta
Explanation
The
eval()method is automatically called when theDiracDeltaclass is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words,eval()method is not needed to be called explicitly, it is being called and evaluated once the object is called.Examples
>>> from sympy import DiracDelta, S >>> from sympy.abc import x
>>> DiracDelta(x) DiracDelta(x)
>>> DiracDelta(-x, 1) -DiracDelta(x, 1)
>>> DiracDelta(1) 0
>>> DiracDelta(5, 1) 0
>>> DiracDelta(0) DiracDelta(0)
>>> DiracDelta(-1) 0
>>> DiracDelta(S.NaN) nan
>>> DiracDelta(x).eval(1) 0
>>> DiracDelta(x - 100).subs(x, 5) 0
>>> DiracDelta(x - 100).subs(x, 100) DiracDelta(0)
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fdiff(argindex=1)[source]¶ Returns the first derivative of a DiracDelta Function.
- Parameters
argindex : integer
degree of derivative
Explanation
The difference between
diff()andfdiff()is:diff()is the user-level function andfdiff()is an object method.fdiff()is a convenience method available in theFunctionclass. It returns the derivative of the function without considering the chain rule.diff(function, x)callsFunction._eval_derivativewhich in turn callsfdiff()internally to compute the derivative of the function.Examples
>>> from sympy import DiracDelta, diff >>> from sympy.abc import x
>>> DiracDelta(x).fdiff() DiracDelta(x, 1)
>>> DiracDelta(x, 1).fdiff() DiracDelta(x, 2)
>>> DiracDelta(x**2 - 1).fdiff() DiracDelta(x**2 - 1, 1)
>>> diff(DiracDelta(x, 1)).fdiff() DiracDelta(x, 3)
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is_simple(x)[source]¶ Tells whether the argument(args[0]) of DiracDelta is a linear expression in x.
- Parameters
x : can be a symbol
Examples
>>> from sympy import DiracDelta, cos >>> from sympy.abc import x, y
>>> DiracDelta(x*y).is_simple(x) True >>> DiracDelta(x*y).is_simple(y) True
>>> DiracDelta(x**2 + x - 2).is_simple(x) False
>>> DiracDelta(cos(x)).is_simple(x) False
See also
Heaviside¶
-
class
sympy.functions.special.delta_functions.Heaviside(arg, H0=None, **options)[source]¶ Heaviside Piecewise function.
Explanation
Heaviside function has the following properties:
\(\frac{d}{d x} \theta(x) = \delta(x)\)
\(\theta(x) = \begin{cases} 0 & \text{for}\: x < 0 \\ \text{undefined} & \text{for}\: x = 0 \\1 & \text{for}\: x > 0 \end{cases}\)
\(\frac{d}{d x} \max(x, 0) = \theta(x)\)
Heaviside(x) is printed as \(\theta(x)\) with the SymPy LaTeX printer.
Regarding to the value at 0, Mathematica defines \(\theta(0)=1\), but Maple uses \(\theta(0) = \text{undefined}\). Different application areas may have specific conventions. For example, in control theory, it is common practice to assume \(\theta(0) = 0\) to match the Laplace transform of a DiracDelta distribution.
To specify the value of Heaviside at
x=0, a second argument can be given. Omit this 2nd argument or passNoneto recover the default behavior.Examples
>>> from sympy import Heaviside, S >>> from sympy.abc import x >>> Heaviside(9) 1 >>> Heaviside(-9) 0 >>> Heaviside(0) Heaviside(0) >>> Heaviside(0, S.Half) 1/2 >>> (Heaviside(x) + 1).replace(Heaviside(x), Heaviside(x, 1)) Heaviside(x, 1) + 1
See also
References
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classmethod
eval(arg, H0=None)[source]¶ Returns a simplified form or a value of Heaviside depending on the argument passed by the Heaviside object.
- Parameters
arg : argument passed by HeaviSide object
HO : boolean flag for HeaviSide Object is set to True
Explanation
The
eval()method is automatically called when theHeavisideclass is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words,eval()method is not needed to be called explicitly, it is being called and evaluated once the object is called.Examples
>>> from sympy import Heaviside, S >>> from sympy.abc import x
>>> Heaviside(x) Heaviside(x)
>>> Heaviside(19) 1
>>> Heaviside(0) Heaviside(0)
>>> Heaviside(0, 1) 1
>>> Heaviside(-5) 0
>>> Heaviside(S.NaN) nan
>>> Heaviside(x).eval(100) 1
>>> Heaviside(x - 100).subs(x, 5) 0
>>> Heaviside(x - 100).subs(x, 105) 1
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fdiff(argindex=1)[source]¶ Returns the first derivative of a Heaviside Function.
- Parameters
argindex : integer
order of derivative
Examples
>>> from sympy import Heaviside, diff >>> from sympy.abc import x
>>> Heaviside(x).fdiff() DiracDelta(x)
>>> Heaviside(x**2 - 1).fdiff() DiracDelta(x**2 - 1)
>>> diff(Heaviside(x)).fdiff() DiracDelta(x, 1)
Singularity Function¶
-
class
sympy.functions.special.singularity_functions.SingularityFunction(**kwargs)[source]¶ Singularity functions are a class of discontinuous functions.
Explanation
Singularity functions take a variable, an offset, and an exponent as arguments. These functions are represented using Macaulay brackets as:
SingularityFunction(x, a, n) := <x - a>^n
The singularity function will automatically evaluate to
Derivative(DiracDelta(x - a), x, -n - 1)ifn < 0and(x - a)**n*Heaviside(x - a)ifn >= 0.Examples
>>> from sympy import SingularityFunction, diff, Piecewise, DiracDelta, Heaviside, Symbol >>> from sympy.abc import x, a, n >>> SingularityFunction(x, a, n) SingularityFunction(x, a, n) >>> y = Symbol('y', positive=True) >>> n = Symbol('n', nonnegative=True) >>> SingularityFunction(y, -10, n) (y + 10)**n >>> y = Symbol('y', negative=True) >>> SingularityFunction(y, 10, n) 0 >>> SingularityFunction(x, 4, -1).subs(x, 4) oo >>> SingularityFunction(x, 10, -2).subs(x, 10) oo >>> SingularityFunction(4, 1, 5) 243 >>> diff(SingularityFunction(x, 1, 5) + SingularityFunction(x, 1, 4), x) 4*SingularityFunction(x, 1, 3) + 5*SingularityFunction(x, 1, 4) >>> diff(SingularityFunction(x, 4, 0), x, 2) SingularityFunction(x, 4, -2) >>> SingularityFunction(x, 4, 5).rewrite(Piecewise) Piecewise(((x - 4)**5, x - 4 > 0), (0, True)) >>> expr = SingularityFunction(x, a, n) >>> y = Symbol('y', positive=True) >>> n = Symbol('n', nonnegative=True) >>> expr.subs({x: y, a: -10, n: n}) (y + 10)**n
The methods
rewrite(DiracDelta),rewrite(Heaviside), andrewrite('HeavisideDiracDelta')returns the same output. One can use any of these methods according to their choice.>>> expr = SingularityFunction(x, 4, 5) + SingularityFunction(x, -3, -1) - SingularityFunction(x, 0, -2) >>> expr.rewrite(Heaviside) (x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1) >>> expr.rewrite(DiracDelta) (x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1) >>> expr.rewrite('HeavisideDiracDelta') (x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)
See also
References
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classmethod
eval(variable, offset, exponent)[source]¶ Returns a simplified form or a value of Singularity Function depending on the argument passed by the object.
Explanation
The
eval()method is automatically called when theSingularityFunctionclass is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words,eval()method is not needed to be called explicitly, it is being called and evaluated once the object is called.Examples
>>> from sympy import SingularityFunction, Symbol, nan >>> from sympy.abc import x, a, n >>> SingularityFunction(x, a, n) SingularityFunction(x, a, n) >>> SingularityFunction(5, 3, 2) 4 >>> SingularityFunction(x, a, nan) nan >>> SingularityFunction(x, 3, 0).subs(x, 3) 1 >>> SingularityFunction(x, a, n).eval(3, 5, 1) 0 >>> SingularityFunction(x, a, n).eval(4, 1, 5) 243 >>> x = Symbol('x', positive = True) >>> a = Symbol('a', negative = True) >>> n = Symbol('n', nonnegative = True) >>> SingularityFunction(x, a, n) (-a + x)**n >>> x = Symbol('x', negative = True) >>> a = Symbol('a', positive = True) >>> SingularityFunction(x, a, n) 0
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fdiff(argindex=1)[source]¶ Returns the first derivative of a DiracDelta Function.
Explanation
The difference between
diff()andfdiff()is:diff()is the user-level function andfdiff()is an object method.fdiff()is a convenience method available in theFunctionclass. It returns the derivative of the function without considering the chain rule.diff(function, x)callsFunction._eval_derivativewhich in turn callsfdiff()internally to compute the derivative of the function.
-
classmethod
Gamma, Beta and related Functions¶
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class
sympy.functions.special.gamma_functions.gamma(**kwargs)[source]¶ The gamma function
\[\Gamma(x) := \int^{\infty}_{0} t^{x-1} e^{-t} \mathrm{d}t.\]Explanation
The
gammafunction implements the function which passes through the values of the factorial function (i.e., \(\Gamma(n) = (n - 1)!\) when n is an integer). More generally, \(\Gamma(z)\) is defined in the whole complex plane except at the negative integers where there are simple poles.Examples
>>> from sympy import S, I, pi, gamma >>> from sympy.abc import x
Several special values are known:
>>> gamma(1) 1 >>> gamma(4) 6 >>> gamma(S(3)/2) sqrt(pi)/2
The
gammafunction obeys the mirror symmetry:>>> from sympy import conjugate >>> conjugate(gamma(x)) gamma(conjugate(x))
Differentiation with respect to \(x\) is supported:
>>> from sympy import diff >>> diff(gamma(x), x) gamma(x)*polygamma(0, x)
Series expansion is also supported:
>>> from sympy import series >>> series(gamma(x), x, 0, 3) 1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 + polygamma(2, 1)/6 - EulerGamma**3/6) + O(x**3)
We can numerically evaluate the
gammafunction to arbitrary precision on the whole complex plane:>>> gamma(pi).evalf(40) 2.288037795340032417959588909060233922890 >>> gamma(1+I).evalf(20) 0.49801566811835604271 - 0.15494982830181068512*I
See also
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
polygammaPolygamma function.
loggammaLog Gamma function.
digammaDigamma function.
trigammaTrigamma function.
betaEuler Beta function.
References
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class
sympy.functions.special.gamma_functions.loggamma(**kwargs)[source]¶ The
loggammafunction implements the logarithm of the gamma function (i.e., \(\log\Gamma(x)\)).Examples
Several special values are known. For numerical integral arguments we have:
>>> from sympy import loggamma >>> loggamma(-2) oo >>> loggamma(0) oo >>> loggamma(1) 0 >>> loggamma(2) 0 >>> loggamma(3) log(2)
And for symbolic values:
>>> from sympy import Symbol >>> n = Symbol("n", integer=True, positive=True) >>> loggamma(n) log(gamma(n)) >>> loggamma(-n) oo
For half-integral values:
>>> from sympy import S >>> loggamma(S(5)/2) log(3*sqrt(pi)/4) >>> loggamma(n/2) log(2**(1 - n)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2))
And general rational arguments:
>>> from sympy import expand_func >>> L = loggamma(S(16)/3) >>> expand_func(L).doit() -5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13) >>> L = loggamma(S(19)/4) >>> expand_func(L).doit() -4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15) >>> L = loggamma(S(23)/7) >>> expand_func(L).doit() -3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16)
The
loggammafunction has the following limits towards infinity:>>> from sympy import oo >>> loggamma(oo) oo >>> loggamma(-oo) zoo
The
loggammafunction obeys the mirror symmetry if \(x \in \mathbb{C} \setminus \{-\infty, 0\}\):>>> from sympy.abc import x >>> from sympy import conjugate >>> conjugate(loggamma(x)) loggamma(conjugate(x))
Differentiation with respect to \(x\) is supported:
>>> from sympy import diff >>> diff(loggamma(x), x) polygamma(0, x)
Series expansion is also supported:
>>> from sympy import series >>> series(loggamma(x), x, 0, 4).cancel() -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + O(x**4)
We can numerically evaluate the
gammafunction to arbitrary precision on the whole complex plane:>>> from sympy import I >>> loggamma(5).evalf(30) 3.17805383034794561964694160130 >>> loggamma(I).evalf(20) -0.65092319930185633889 - 1.8724366472624298171*I
See also
gammaGamma function.
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
polygammaPolygamma function.
digammaDigamma function.
trigammaTrigamma function.
betaEuler Beta function.
References
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class
sympy.functions.special.gamma_functions.polygamma(**kwargs)[source]¶ The function
polygamma(n, z)returnslog(gamma(z)).diff(n + 1).Explanation
It is a meromorphic function on \(\mathbb{C}\) and defined as the \((n+1)\)-th derivative of the logarithm of the gamma function:
\[\psi^{(n)} (z) := \frac{\mathrm{d}^{n+1}}{\mathrm{d} z^{n+1}} \log\Gamma(z).\]Examples
Several special values are known:
>>> from sympy import S, polygamma >>> polygamma(0, 1) -EulerGamma >>> polygamma(0, 1/S(2)) -2*log(2) - EulerGamma >>> polygamma(0, 1/S(3)) -log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3)) >>> polygamma(0, 1/S(4)) -pi/2 - log(4) - log(2) - EulerGamma >>> polygamma(0, 2) 1 - EulerGamma >>> polygamma(0, 23) 19093197/5173168 - EulerGamma
>>> from sympy import oo, I >>> polygamma(0, oo) oo >>> polygamma(0, -oo) oo >>> polygamma(0, I*oo) oo >>> polygamma(0, -I*oo) oo
Differentiation with respect to \(x\) is supported:
>>> from sympy import Symbol, diff >>> x = Symbol("x") >>> diff(polygamma(0, x), x) polygamma(1, x) >>> diff(polygamma(0, x), x, 2) polygamma(2, x) >>> diff(polygamma(0, x), x, 3) polygamma(3, x) >>> diff(polygamma(1, x), x) polygamma(2, x) >>> diff(polygamma(1, x), x, 2) polygamma(3, x) >>> diff(polygamma(2, x), x) polygamma(3, x) >>> diff(polygamma(2, x), x, 2) polygamma(4, x)
>>> n = Symbol("n") >>> diff(polygamma(n, x), x) polygamma(n + 1, x) >>> diff(polygamma(n, x), x, 2) polygamma(n + 2, x)
We can rewrite
polygammafunctions in terms of harmonic numbers:>>> from sympy import harmonic >>> polygamma(0, x).rewrite(harmonic) harmonic(x - 1) - EulerGamma >>> polygamma(2, x).rewrite(harmonic) 2*harmonic(x - 1, 3) - 2*zeta(3) >>> ni = Symbol("n", integer=True) >>> polygamma(ni, x).rewrite(harmonic) (-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n)
See also
gammaGamma function.
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
loggammaLog Gamma function.
digammaDigamma function.
trigammaTrigamma function.
betaEuler Beta function.
References
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class
sympy.functions.special.gamma_functions.digamma(**kwargs)[source]¶ The
digammafunction is the first derivative of theloggammafunction\[\psi(x) := \frac{\mathrm{d}}{\mathrm{d} z} \log\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z) }.\]In this case,
digamma(z) = polygamma(0, z).Examples
>>> from sympy import digamma >>> digamma(0) zoo >>> from sympy import Symbol >>> z = Symbol('z') >>> digamma(z) polygamma(0, z)
To retain
digammaas it is:>>> digamma(0, evaluate=False) digamma(0) >>> digamma(z, evaluate=False) digamma(z)
See also
gammaGamma function.
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
polygammaPolygamma function.
loggammaLog Gamma function.
trigammaTrigamma function.
betaEuler Beta function.
References
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class
sympy.functions.special.gamma_functions.trigamma(**kwargs)[source]¶ The
trigammafunction is the second derivative of theloggammafunction\[\psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z).\]In this case,
trigamma(z) = polygamma(1, z).Examples
>>> from sympy import trigamma >>> trigamma(0) zoo >>> from sympy import Symbol >>> z = Symbol('z') >>> trigamma(z) polygamma(1, z)
To retain
trigammaas it is:>>> trigamma(0, evaluate=False) trigamma(0) >>> trigamma(z, evaluate=False) trigamma(z)
See also
gammaGamma function.
lowergammaLower incomplete gamma function.
uppergammaUpper incomplete gamma function.
polygammaPolygamma function.
loggammaLog Gamma function.
digammaDigamma function.
betaEuler Beta function.
References
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class
sympy.functions.special.gamma_functions.uppergamma(**kwargs)[source]¶ The upper incomplete gamma function.
Explanation
It can be defined as the meromorphic continuation of
\[\Gamma(s, x) := \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \gamma(s, x).\]where \(\gamma(s, x)\) is the lower incomplete gamma function,
lowergamma. This can be shown to be the same as\[\Gamma(s, x) = \Gamma(s) - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]where \({}_1F_1\) is the (confluent) hypergeometric function.
The upper incomplete gamma function is also essentially equivalent to the generalized exponential integral:
\[\operatorname{E}_{n}(x) = \int_{1}^{\infty}{\frac{e^{-xt}}{t^n} \, dt} = x^{n-1}\Gamma(1-n,x).\]Examples
>>> from sympy import uppergamma, S >>> from sympy.abc import s, x >>> uppergamma(s, x) uppergamma(s, x) >>> uppergamma(3, x) 2*(x**2/2 + x + 1)*exp(-x) >>> uppergamma(-S(1)/2, x) -2*sqrt(pi)*erfc(sqrt(x)) + 2*exp(-x)/sqrt(x) >>> uppergamma(-2, x) expint(3, x)/x**2
See also
References
- R306
https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function
- R307
Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
- R308
- R309
- R310
- R311
https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions
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class
sympy.functions.special.gamma_functions.lowergamma(**kwargs)[source]¶ The lower incomplete gamma function.
Explanation
It can be defined as the meromorphic continuation of
\[\gamma(s, x) := \int_0^x t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \Gamma(s, x).\]This can be shown to be the same as
\[\gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]where \({}_1F_1\) is the (confluent) hypergeometric function.
Examples
>>> from sympy import lowergamma, S >>> from sympy.abc import s, x >>> lowergamma(s, x) lowergamma(s, x) >>> lowergamma(3, x) -2*(x**2/2 + x + 1)*exp(-x) + 2 >>> lowergamma(-S(1)/2, x) -2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x)
See also
References
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class
sympy.functions.special.gamma_functions.multigamma(**kwargs)[source]¶ The multivariate gamma function is a generalization of the gamma function
\[\Gamma_p(z) = \pi^{p(p-1)/4}\prod_{k=1}^p \Gamma[z + (1 - k)/2].\]In a special case,
multigamma(x, 1) = gamma(x).- Parameters
p : order or dimension of the multivariate gamma function
Examples
>>> from sympy import S, multigamma >>> from sympy import Symbol >>> x = Symbol('x') >>> p = Symbol('p', positive=True, integer=True)
>>> multigamma(x, p) pi**(p*(p - 1)/4)*Product(gamma(-_k/2 + x + 1/2), (_k, 1, p))
Several special values are known:
>>> multigamma(1, 1) 1 >>> multigamma(4, 1) 6 >>> multigamma(S(3)/2, 1) sqrt(pi)/2
Writing
multigammain terms of thegammafunction:>>> multigamma(x, 1) gamma(x)
>>> multigamma(x, 2) sqrt(pi)*gamma(x)*gamma(x - 1/2)
>>> multigamma(x, 3) pi**(3/2)*gamma(x)*gamma(x - 1)*gamma(x - 1/2)
See also
gamma,lowergamma,uppergamma,polygamma,loggamma,digamma,trigamma,betaReferences
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class
sympy.functions.special.beta_functions.beta(**kwargs)[source]¶ The beta integral is called the Eulerian integral of the first kind by Legendre:
\[\mathrm{B}(x,y) \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t.\]Explanation
The Beta function or Euler’s first integral is closely associated with the gamma function. The Beta function is often used in probability theory and mathematical statistics. It satisfies properties like:
\[\begin{split}\mathrm{B}(a,1) = \frac{1}{a} \\ \mathrm{B}(a,b) = \mathrm{B}(b,a) \\ \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\end{split}\]Therefore for integral values of \(a\) and \(b\):
\[\mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!}\]A special case of the Beta function when \(x = y\) is the Central Beta function. It satisfies properties like:
\[\mathrm{B}(x) = 2^{1 - 2x}\mathrm{B}(x, \frac{1}{2}) \mathrm{B}(x) = 2^{1 - 2x} cos(\pi x) \mathrm{B}(\frac{1}{2} - x, x) \mathrm{B}(x) = \int_{0}^{1} \frac{t^x}{(1 + t)^{2x}} dt \mathrm{B}(x) = \frac{2}{x} \prod_{n = 1}^{\infty} \frac{n(n + 2x)}{(n + x)^2}\]Examples
>>> from sympy import I, pi >>> from sympy.abc import x, y
The Beta function obeys the mirror symmetry:
>>> from sympy import beta, conjugate >>> conjugate(beta(x, y)) beta(conjugate(x), conjugate(y))
Differentiation with respect to both \(x\) and \(y\) is supported:
>>> from sympy import beta, diff >>> diff(beta(x, y), x) (polygamma(0, x) - polygamma(0, x + y))*beta(x, y)
>>> diff(beta(x, y), y) (polygamma(0, y) - polygamma(0, x + y))*beta(x, y)
>>> diff(beta(x), x) 2*(polygamma(0, x) - polygamma(0, 2*x))*beta(x, x)
We can numerically evaluate the Beta function to arbitrary precision for any complex numbers x and y:
>>> from sympy import beta >>> beta(pi).evalf(40) 0.02671848900111377452242355235388489324562
>>> beta(1 + I).evalf(20) -0.2112723729365330143 - 0.7655283165378005676*I
See also
gammaGamma function.
uppergammaUpper incomplete gamma function.
lowergammaLower incomplete gamma function.
polygammaPolygamma function.
loggammaLog Gamma function.
digammaDigamma function.
trigammaTrigamma function.
References
Error Functions and Fresnel Integrals¶
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class
sympy.functions.special.error_functions.erf(**kwargs)[source]¶ The Gauss error function.
Explanation
This function is defined as:
\[\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{d}t.\]Examples
>>> from sympy import I, oo, erf >>> from sympy.abc import z
Several special values are known:
>>> erf(0) 0 >>> erf(oo) 1 >>> erf(-oo) -1 >>> erf(I*oo) oo*I >>> erf(-I*oo) -oo*I
In general one can pull out factors of -1 and \(I\) from the argument:
>>> erf(-z) -erf(z)
The error function obeys the mirror symmetry:
>>> from sympy import conjugate >>> conjugate(erf(z)) erf(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(erf(z), z) 2*exp(-z**2)/sqrt(pi)
We can numerically evaluate the error function to arbitrary precision on the whole complex plane:
>>> erf(4).evalf(30) 0.999999984582742099719981147840
>>> erf(-4*I).evalf(30) -1296959.73071763923152794095062*I
See also
References
-
class
sympy.functions.special.error_functions.erfc(**kwargs)[source]¶ Complementary Error Function.
Explanation
The function is defined as:
\[\mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \mathrm{d}t\]Examples
>>> from sympy import I, oo, erfc >>> from sympy.abc import z
Several special values are known:
>>> erfc(0) 1 >>> erfc(oo) 0 >>> erfc(-oo) 2 >>> erfc(I*oo) -oo*I >>> erfc(-I*oo) oo*I
The error function obeys the mirror symmetry:
>>> from sympy import conjugate >>> conjugate(erfc(z)) erfc(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(erfc(z), z) -2*exp(-z**2)/sqrt(pi)
It also follows
>>> erfc(-z) 2 - erfc(z)
We can numerically evaluate the complementary error function to arbitrary precision on the whole complex plane:
>>> erfc(4).evalf(30) 0.0000000154172579002800188521596734869
>>> erfc(4*I).evalf(30) 1.0 - 1296959.73071763923152794095062*I
See also
References
-
class
sympy.functions.special.error_functions.erfi(**kwargs)[source]¶ Imaginary error function.
Explanation
The function erfi is defined as:
\[\mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} \mathrm{d}t\]Examples
>>> from sympy import I, oo, erfi >>> from sympy.abc import z
Several special values are known:
>>> erfi(0) 0 >>> erfi(oo) oo >>> erfi(-oo) -oo >>> erfi(I*oo) I >>> erfi(-I*oo) -I
In general one can pull out factors of -1 and \(I\) from the argument:
>>> erfi(-z) -erfi(z)
>>> from sympy import conjugate >>> conjugate(erfi(z)) erfi(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(erfi(z), z) 2*exp(z**2)/sqrt(pi)
We can numerically evaluate the imaginary error function to arbitrary precision on the whole complex plane:
>>> erfi(2).evalf(30) 18.5648024145755525987042919132
>>> erfi(-2*I).evalf(30) -0.995322265018952734162069256367*I
See also
References
-
class
sympy.functions.special.error_functions.erf2(**kwargs)[source]¶ Two-argument error function.
Explanation
This function is defined as:
\[\mathrm{erf2}(x, y) = \frac{2}{\sqrt{\pi}} \int_x^y e^{-t^2} \mathrm{d}t\]Examples
>>> from sympy import oo, erf2 >>> from sympy.abc import x, y
Several special values are known:
>>> erf2(0, 0) 0 >>> erf2(x, x) 0 >>> erf2(x, oo) 1 - erf(x) >>> erf2(x, -oo) -erf(x) - 1 >>> erf2(oo, y) erf(y) - 1 >>> erf2(-oo, y) erf(y) + 1
In general one can pull out factors of -1:
>>> erf2(-x, -y) -erf2(x, y)
The error function obeys the mirror symmetry:
>>> from sympy import conjugate >>> conjugate(erf2(x, y)) erf2(conjugate(x), conjugate(y))
Differentiation with respect to \(x\), \(y\) is supported:
>>> from sympy import diff >>> diff(erf2(x, y), x) -2*exp(-x**2)/sqrt(pi) >>> diff(erf2(x, y), y) 2*exp(-y**2)/sqrt(pi)
See also
References
-
class
sympy.functions.special.error_functions.erfinv(**kwargs)[source]¶ Inverse Error Function. The erfinv function is defined as:
\[\mathrm{erf}(x) = y \quad \Rightarrow \quad \mathrm{erfinv}(y) = x\]Examples
>>> from sympy import erfinv >>> from sympy.abc import x
Several special values are known:
>>> erfinv(0) 0 >>> erfinv(1) oo
Differentiation with respect to \(x\) is supported:
>>> from sympy import diff >>> diff(erfinv(x), x) sqrt(pi)*exp(erfinv(x)**2)/2
We can numerically evaluate the inverse error function to arbitrary precision on [-1, 1]:
>>> erfinv(0.2).evalf(30) 0.179143454621291692285822705344
See also
References
-
class
sympy.functions.special.error_functions.erfcinv(**kwargs)[source]¶ Inverse Complementary Error Function. The erfcinv function is defined as:
\[\mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x\]Examples
>>> from sympy import erfcinv >>> from sympy.abc import x
Several special values are known:
>>> erfcinv(1) 0 >>> erfcinv(0) oo
Differentiation with respect to \(x\) is supported:
>>> from sympy import diff >>> diff(erfcinv(x), x) -sqrt(pi)*exp(erfcinv(x)**2)/2
See also
References
-
class
sympy.functions.special.error_functions.erf2inv(**kwargs)[source]¶ Two-argument Inverse error function. The erf2inv function is defined as:
\[\mathrm{erf2}(x, w) = y \quad \Rightarrow \quad \mathrm{erf2inv}(x, y) = w\]Examples
>>> from sympy import erf2inv, oo >>> from sympy.abc import x, y
Several special values are known:
>>> erf2inv(0, 0) 0 >>> erf2inv(1, 0) 1 >>> erf2inv(0, 1) oo >>> erf2inv(0, y) erfinv(y) >>> erf2inv(oo, y) erfcinv(-y)
Differentiation with respect to \(x\) and \(y\) is supported:
>>> from sympy import diff >>> diff(erf2inv(x, y), x) exp(-x**2 + erf2inv(x, y)**2) >>> diff(erf2inv(x, y), y) sqrt(pi)*exp(erf2inv(x, y)**2)/2
See also
References
-
class
sympy.functions.special.error_functions.FresnelIntegral(**kwargs)[source]¶ Base class for the Fresnel integrals.
-
class
sympy.functions.special.error_functions.fresnels(**kwargs)[source]¶ Fresnel integral S.
Explanation
This function is defined by
\[\operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.\]It is an entire function.
Examples
>>> from sympy import I, oo, fresnels >>> from sympy.abc import z
Several special values are known:
>>> fresnels(0) 0 >>> fresnels(oo) 1/2 >>> fresnels(-oo) -1/2 >>> fresnels(I*oo) -I/2 >>> fresnels(-I*oo) I/2
In general one can pull out factors of -1 and \(i\) from the argument:
>>> fresnels(-z) -fresnels(z) >>> fresnels(I*z) -I*fresnels(z)
The Fresnel S integral obeys the mirror symmetry \(\overline{S(z)} = S(\bar{z})\):
>>> from sympy import conjugate >>> conjugate(fresnels(z)) fresnels(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(fresnels(z), z) sin(pi*z**2/2)
Defining the Fresnel functions via an integral:
>>> from sympy import integrate, pi, sin, expand_func >>> integrate(sin(pi*z**2/2), z) 3*fresnels(z)*gamma(3/4)/(4*gamma(7/4)) >>> expand_func(integrate(sin(pi*z**2/2), z)) fresnels(z)
We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane:
>>> fresnels(2).evalf(30) 0.343415678363698242195300815958
>>> fresnels(-2*I).evalf(30) 0.343415678363698242195300815958*I
See also
fresnelcFresnel cosine integral.
References
-
class
sympy.functions.special.error_functions.fresnelc(**kwargs)[source]¶ Fresnel integral C.
Explanation
This function is defined by
\[\operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.\]It is an entire function.
Examples
>>> from sympy import I, oo, fresnelc >>> from sympy.abc import z
Several special values are known:
>>> fresnelc(0) 0 >>> fresnelc(oo) 1/2 >>> fresnelc(-oo) -1/2 >>> fresnelc(I*oo) I/2 >>> fresnelc(-I*oo) -I/2
In general one can pull out factors of -1 and \(i\) from the argument:
>>> fresnelc(-z) -fresnelc(z) >>> fresnelc(I*z) I*fresnelc(z)
The Fresnel C integral obeys the mirror symmetry \(\overline{C(z)} = C(\bar{z})\):
>>> from sympy import conjugate >>> conjugate(fresnelc(z)) fresnelc(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(fresnelc(z), z) cos(pi*z**2/2)
Defining the Fresnel functions via an integral:
>>> from sympy import integrate, pi, cos, expand_func >>> integrate(cos(pi*z**2/2), z) fresnelc(z)*gamma(1/4)/(4*gamma(5/4)) >>> expand_func(integrate(cos(pi*z**2/2), z)) fresnelc(z)
We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane:
>>> fresnelc(2).evalf(30) 0.488253406075340754500223503357
>>> fresnelc(-2*I).evalf(30) -0.488253406075340754500223503357*I
See also
fresnelsFresnel sine integral.
References
Exponential, Logarithmic and Trigonometric Integrals¶
-
class
sympy.functions.special.error_functions.Ei(**kwargs)[source]¶ The classical exponential integral.
Explanation
For use in SymPy, this function is defined as
\[\operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!} + \log(x) + \gamma,\]where \(\gamma\) is the Euler-Mascheroni constant.
If \(x\) is a polar number, this defines an analytic function on the Riemann surface of the logarithm. Otherwise this defines an analytic function in the cut plane \(\mathbb{C} \setminus (-\infty, 0]\).
Background
The name exponential integral comes from the following statement:
\[\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t\]If the integral is interpreted as a Cauchy principal value, this statement holds for \(x > 0\) and \(\operatorname{Ei}(x)\) as defined above.
Examples
>>> from sympy import Ei, polar_lift, exp_polar, I, pi >>> from sympy.abc import x
>>> Ei(-1) Ei(-1)
This yields a real value:
>>> Ei(-1).n(chop=True) -0.219383934395520
On the other hand the analytic continuation is not real:
>>> Ei(polar_lift(-1)).n(chop=True) -0.21938393439552 + 3.14159265358979*I
The exponential integral has a logarithmic branch point at the origin:
>>> Ei(x*exp_polar(2*I*pi)) Ei(x) + 2*I*pi
Differentiation is supported:
>>> Ei(x).diff(x) exp(x)/x
The exponential integral is related to many other special functions. For example:
>>> from sympy import expint, Shi >>> Ei(x).rewrite(expint) -expint(1, x*exp_polar(I*pi)) - I*pi >>> Ei(x).rewrite(Shi) Chi(x) + Shi(x)
See also
References
- R348
- R349
- R350
Abramowitz & Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm
-
class
sympy.functions.special.error_functions.expint(**kwargs)[source]¶ Generalized exponential integral.
Explanation
This function is defined as
\[\operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),\]where \(\Gamma(1 - \nu, z)\) is the upper incomplete gamma function (
uppergamma).Hence for \(z\) with positive real part we have
\[\operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t,\]which explains the name.
The representation as an incomplete gamma function provides an analytic continuation for \(\operatorname{E}_\nu(z)\). If \(\nu\) is a non-positive integer, the exponential integral is thus an unbranched function of \(z\), otherwise there is a branch point at the origin. Refer to the incomplete gamma function documentation for details of the branching behavior.
Examples
>>> from sympy import expint, S >>> from sympy.abc import nu, z
Differentiation is supported. Differentiation with respect to \(z\) further explains the name: for integral orders, the exponential integral is an iterated integral of the exponential function.
>>> expint(nu, z).diff(z) -expint(nu - 1, z)
Differentiation with respect to \(\nu\) has no classical expression:
>>> expint(nu, z).diff(nu) -z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, 1 - nu), ()), z)
At non-postive integer orders, the exponential integral reduces to the exponential function:
>>> expint(0, z) exp(-z)/z >>> expint(-1, z) exp(-z)/z + exp(-z)/z**2
At half-integers it reduces to error functions:
>>> expint(S(1)/2, z) sqrt(pi)*erfc(sqrt(z))/sqrt(z)
At positive integer orders it can be rewritten in terms of exponentials and
expint(1, z). Useexpand_func()to do this:>>> from sympy import expand_func >>> expand_func(expint(5, z)) z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24
The generalised exponential integral is essentially equivalent to the incomplete gamma function:
>>> from sympy import uppergamma >>> expint(nu, z).rewrite(uppergamma) z**(nu - 1)*uppergamma(1 - nu, z)
As such it is branched at the origin:
>>> from sympy import exp_polar, pi, I >>> expint(4, z*exp_polar(2*pi*I)) I*pi*z**3/3 + expint(4, z) >>> expint(nu, z*exp_polar(2*pi*I)) z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(1 - nu) + expint(nu, z)
See also
References
-
sympy.functions.special.error_functions.E1(z)[source]¶ Classical case of the generalized exponential integral.
Explanation
This is equivalent to
expint(1, z).Examples
>>> from sympy import E1 >>> E1(0) expint(1, 0)
>>> E1(5) expint(1, 5)
-
class
sympy.functions.special.error_functions.li(**kwargs)[source]¶ The classical logarithmic integral.
Explanation
For use in SymPy, this function is defined as
\[\operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.\]Examples
>>> from sympy import I, oo, li >>> from sympy.abc import z
Several special values are known:
>>> li(0) 0 >>> li(1) -oo >>> li(oo) oo
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(li(z), z) 1/log(z)
Defining the
lifunction via an integral: >>> from sympy import integrate >>> integrate(li(z)) z*li(z) - Ei(2*log(z))>>> integrate(li(z),z) z*li(z) - Ei(2*log(z))
The logarithmic integral can also be defined in terms of
Ei:>>> from sympy import Ei >>> li(z).rewrite(Ei) Ei(log(z)) >>> diff(li(z).rewrite(Ei), z) 1/log(z)
We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points):
>>> li(2).evalf(30) 1.04516378011749278484458888919
>>> li(2*I).evalf(30) 1.0652795784357498247001125598 + 3.08346052231061726610939702133*I
We can even compute Soldner’s constant by the help of mpmath:
>>> from mpmath import findroot >>> findroot(li, 2) 1.45136923488338
Further transformations include rewriting
liin terms of the trigonometric integralsSi,Ci,ShiandChi:>>> from sympy import Si, Ci, Shi, Chi >>> li(z).rewrite(Si) -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)) >>> li(z).rewrite(Ci) -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)) >>> li(z).rewrite(Shi) -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z)) >>> li(z).rewrite(Chi) -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))
See also
References
-
class
sympy.functions.special.error_functions.Li(**kwargs)[source]¶ The offset logarithmic integral.
Explanation
For use in SymPy, this function is defined as
\[\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)\]Examples
>>> from sympy import Li >>> from sympy.abc import z
The following special value is known:
>>> Li(2) 0
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(Li(z), z) 1/log(z)
The shifted logarithmic integral can be written in terms of \(li(z)\):
>>> from sympy import li >>> Li(z).rewrite(li) li(z) - li(2)
We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points):
>>> Li(2).evalf(30) 0
>>> Li(4).evalf(30) 1.92242131492155809316615998938
See also
References
-
class
sympy.functions.special.error_functions.Si(**kwargs)[source]¶ Sine integral.
Explanation
This function is defined by
\[\operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.\]It is an entire function.
Examples
>>> from sympy import Si >>> from sympy.abc import z
The sine integral is an antiderivative of \(sin(z)/z\):
>>> Si(z).diff(z) sin(z)/z
It is unbranched:
>>> from sympy import exp_polar, I, pi >>> Si(z*exp_polar(2*I*pi)) Si(z)
Sine integral behaves much like ordinary sine under multiplication by
I:>>> Si(I*z) I*Shi(z) >>> Si(-z) -Si(z)
It can also be expressed in terms of exponential integrals, but beware that the latter is branched:
>>> from sympy import expint >>> Si(z).rewrite(expint) -I*(-expint(1, z*exp_polar(-I*pi/2))/2 + expint(1, z*exp_polar(I*pi/2))/2) + pi/2
It can be rewritten in the form of sinc function (by definition):
>>> from sympy import sinc >>> Si(z).rewrite(sinc) Integral(sinc(t), (t, 0, z))
See also
References
-
class
sympy.functions.special.error_functions.Ci(**kwargs)[source]¶ Cosine integral.
Explanation
This function is defined for positive \(x\) by
\[\operatorname{Ci}(x) = \gamma + \log{x} + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,\]where \(\gamma\) is the Euler-Mascheroni constant.
We have
\[\operatorname{Ci}(z) = -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right) + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}\]which holds for all polar \(z\) and thus provides an analytic continuation to the Riemann surface of the logarithm.
The formula also holds as stated for \(z \in \mathbb{C}\) with \(\Re(z) > 0\). By lifting to the principal branch, we obtain an analytic function on the cut complex plane.
Examples
>>> from sympy import Ci >>> from sympy.abc import z
The cosine integral is a primitive of \(\cos(z)/z\):
>>> Ci(z).diff(z) cos(z)/z
It has a logarithmic branch point at the origin:
>>> from sympy import exp_polar, I, pi >>> Ci(z*exp_polar(2*I*pi)) Ci(z) + 2*I*pi
The cosine integral behaves somewhat like ordinary \(\cos\) under multiplication by \(i\):
>>> from sympy import polar_lift >>> Ci(polar_lift(I)*z) Chi(z) + I*pi/2 >>> Ci(polar_lift(-1)*z) Ci(z) + I*pi
It can also be expressed in terms of exponential integrals:
>>> from sympy import expint >>> Ci(z).rewrite(expint) -expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2
See also
References
-
class
sympy.functions.special.error_functions.Shi(**kwargs)[source]¶ Sinh integral.
Explanation
This function is defined by
\[\operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.\]It is an entire function.
Examples
>>> from sympy import Shi >>> from sympy.abc import z
The Sinh integral is a primitive of \(\sinh(z)/z\):
>>> Shi(z).diff(z) sinh(z)/z
It is unbranched:
>>> from sympy import exp_polar, I, pi >>> Shi(z*exp_polar(2*I*pi)) Shi(z)
The \(\sinh\) integral behaves much like ordinary \(\sinh\) under multiplication by \(i\):
>>> Shi(I*z) I*Si(z) >>> Shi(-z) -Shi(z)
It can also be expressed in terms of exponential integrals, but beware that the latter is branched:
>>> from sympy import expint >>> Shi(z).rewrite(expint) expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2
See also
References
-
class
sympy.functions.special.error_functions.Chi(**kwargs)[source]¶ Cosh integral.
Explanation
This function is defined for positive \(x\) by
\[\operatorname{Chi}(x) = \gamma + \log{x} + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,\]where \(\gamma\) is the Euler-Mascheroni constant.
We have
\[\operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right) - i\frac{\pi}{2},\]which holds for all polar \(z\) and thus provides an analytic continuation to the Riemann surface of the logarithm. By lifting to the principal branch we obtain an analytic function on the cut complex plane.
Examples
>>> from sympy import Chi >>> from sympy.abc import z
The \(\cosh\) integral is a primitive of \(\cosh(z)/z\):
>>> Chi(z).diff(z) cosh(z)/z
It has a logarithmic branch point at the origin:
>>> from sympy import exp_polar, I, pi >>> Chi(z*exp_polar(2*I*pi)) Chi(z) + 2*I*pi
The \(\cosh\) integral behaves somewhat like ordinary \(\cosh\) under multiplication by \(i\):
>>> from sympy import polar_lift >>> Chi(polar_lift(I)*z) Ci(z) + I*pi/2 >>> Chi(polar_lift(-1)*z) Chi(z) + I*pi
It can also be expressed in terms of exponential integrals:
>>> from sympy import expint >>> Chi(z).rewrite(expint) -expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2
See also
References
Bessel Type Functions¶
-
class
sympy.functions.special.bessel.BesselBase(**kwargs)[source]¶ Abstract base class for Bessel-type functions.
This class is meant to reduce code duplication. All Bessel-type functions can 1) be differentiated, with the derivatives expressed in terms of similar functions, and 2) be rewritten in terms of other Bessel-type functions.
Here, Bessel-type functions are assumed to have one complex parameter.
To use this base class, define class attributes
_aand_bsuch that2*F_n' = -_a*F_{n+1} + b*F_{n-1}.-
property
argument¶ The argument of the Bessel-type function.
-
property
order¶ The order of the Bessel-type function.
-
property
-
class
sympy.functions.special.bessel.besselj(**kwargs)[source]¶ Bessel function of the first kind.
Explanation
The Bessel \(J\) function of order \(\nu\) is defined to be the function satisfying Bessel’s differential equation
\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0,\]with Laurent expansion
\[J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right),\]if \(\nu\) is not a negative integer. If \(\nu=-n \in \mathbb{Z}_{<0}\) is a negative integer, then the definition is
\[J_{-n}(z) = (-1)^n J_n(z).\]Examples
Create a Bessel function object:
>>> from sympy import besselj, jn >>> from sympy.abc import z, n >>> b = besselj(n, z)
Differentiate it:
>>> b.diff(z) besselj(n - 1, z)/2 - besselj(n + 1, z)/2
Rewrite in terms of spherical Bessel functions:
>>> b.rewrite(jn) sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi)
Access the parameter and argument:
>>> b.order n >>> b.argument z
References
-
class
sympy.functions.special.bessel.bessely(**kwargs)[source]¶ Bessel function of the second kind.
Explanation
The Bessel \(Y\) function of order \(\nu\) is defined as
\[Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)},\]where \(J_\mu(z)\) is the Bessel function of the first kind.
It is a solution to Bessel’s equation, and linearly independent from \(J_\nu\).
Examples
>>> from sympy import bessely, yn >>> from sympy.abc import z, n >>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi)
References
-
class
sympy.functions.special.bessel.besseli(**kwargs)[source]¶ Modified Bessel function of the first kind.
Explanation
The Bessel \(I\) function is a solution to the modified Bessel equation
\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0.\]It can be defined as
\[I_\nu(z) = i^{-\nu} J_\nu(iz),\]where \(J_\nu(z)\) is the Bessel function of the first kind.
Examples
>>> from sympy import besseli >>> from sympy.abc import z, n >>> besseli(n, z).diff(z) besseli(n - 1, z)/2 + besseli(n + 1, z)/2
References
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class
sympy.functions.special.bessel.besselk(**kwargs)[source]¶ Modified Bessel function of the second kind.
Explanation
The Bessel \(K\) function of order \(\nu\) is defined as
\[K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)},\]where \(I_\mu(z)\) is the modified Bessel function of the first kind.
It is a solution of the modified Bessel equation, and linearly independent from \(Y_\nu\).
Examples
>>> from sympy import besselk >>> from sympy.abc import z, n >>> besselk(n, z).diff(z) -besselk(n - 1, z)/2 - besselk(n + 1, z)/2
References
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class
sympy.functions.special.bessel.hankel1(**kwargs)[source]¶ Hankel function of the first kind.
Explanation
This function is defined as
\[H_\nu^{(1)} = J_\nu(z) + iY_\nu(z),\]where \(J_\nu(z)\) is the Bessel function of the first kind, and \(Y_\nu(z)\) is the Bessel function of the second kind.
It is a solution to Bessel’s equation.
Examples
>>> from sympy import hankel1 >>> from sympy.abc import z, n >>> hankel1(n, z).diff(z) hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2
References
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class
sympy.functions.special.bessel.hankel2(**kwargs)[source]¶ Hankel function of the second kind.
Explanation
This function is defined as
\[H_\nu^{(2)} = J_\nu(z) - iY_\nu(z),\]where \(J_\nu(z)\) is the Bessel function of the first kind, and \(Y_\nu(z)\) is the Bessel function of the second kind.
It is a solution to Bessel’s equation, and linearly independent from \(H_\nu^{(1)}\).
Examples
>>> from sympy import hankel2 >>> from sympy.abc import z, n >>> hankel2(n, z).diff(z) hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2
References
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class
sympy.functions.special.bessel.jn(**kwargs)[source]¶ Spherical Bessel function of the first kind.
Explanation
This function is a solution to the spherical Bessel equation
\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0.\]It can be defined as
\[j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z),\]where \(J_\nu(z)\) is the Bessel function of the first kind.
The spherical Bessel functions of integral order are calculated using the formula:
\[j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},\]where the coefficients \(f_n(z)\) are available as
sympy.polys.orthopolys.spherical_bessel_fn().Examples
>>> from sympy import Symbol, jn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(jn(0, z))) sin(z)/z >>> expand_func(jn(1, z)) == sin(z)/z**2 - cos(z)/z True >>> expand_func(jn(3, z)) (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z) >>> jn(nu, z).rewrite(besselj) sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2 >>> jn(nu, z).rewrite(bessely) (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2 >>> jn(2, 5.2+0.3j).evalf(20) 0.099419756723640344491 - 0.054525080242173562897*I
References
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class
sympy.functions.special.bessel.yn(**kwargs)[source]¶ Spherical Bessel function of the second kind.
Explanation
This function is another solution to the spherical Bessel equation, and linearly independent from \(j_n\). It can be defined as
\[y_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z),\]where \(Y_\nu(z)\) is the Bessel function of the second kind.
For integral orders \(n\), \(y_n\) is calculated using the formula:
\[y_n(z) = (-1)^{n+1} j_{-n-1}(z)\]Examples
>>> from sympy import Symbol, yn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(yn(0, z))) -cos(z)/z >>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z True >>> yn(nu, z).rewrite(besselj) (-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2 >>> yn(nu, z).rewrite(bessely) sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2 >>> yn(2, 5.2+0.3j).evalf(20) 0.18525034196069722536 + 0.014895573969924817587*I
References
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sympy.functions.special.bessel.jn_zeros(n, k, method='sympy', dps=15)[source]¶ Zeros of the spherical Bessel function of the first kind.
- Parameters
n : integer
order of Bessel function
k : integer
number of zeros to return
Explanation
This returns an array of zeros of \(jn\) up to the \(k\)-th zero.
method = “sympy”: uses mpmath.besseljzero
method = “scipy”: uses the SciPy’s sph_jn and newton to find all roots, which is faster than computing the zeros using a general numerical solver, but it requires SciPy and only works with low precision floating point numbers. (The function used with method=”sympy” is a recent addition to mpmath; before that a general solver was used.)
Examples
>>> from sympy import jn_zeros >>> jn_zeros(2, 4, dps=5) [5.7635, 9.095, 12.323, 15.515]
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class
sympy.functions.special.bessel.marcumq(**kwargs)[source]¶ The Marcum Q-function.
Explanation
The Marcum Q-function is defined by the meromorphic continuation of
\[Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx\]Examples
>>> from sympy import marcumq >>> from sympy.abc import m, a, b >>> marcumq(m, a, b) marcumq(m, a, b)
Special values:
>>> marcumq(m, 0, b) uppergamma(m, b**2/2)/gamma(m) >>> marcumq(0, 0, 0) 0 >>> marcumq(0, a, 0) 1 - exp(-a**2/2) >>> marcumq(1, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 >>> marcumq(2, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2)
Differentiation with respect to \(a\) and \(b\) is supported:
>>> from sympy import diff >>> diff(marcumq(m, a, b), a) a*(-marcumq(m, a, b) + marcumq(m + 1, a, b)) >>> diff(marcumq(m, a, b), b) -a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b)
References
Airy Functions¶
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class
sympy.functions.special.bessel.AiryBase(**kwargs)[source]¶ Abstract base class for Airy functions.
This class is meant to reduce code duplication.
-
class
sympy.functions.special.bessel.airyai(**kwargs)[source]¶ The Airy function \(\operatorname{Ai}\) of the first kind.
Explanation
The Airy function \(\operatorname{Ai}(z)\) is defined to be the function satisfying Airy’s differential equation
\[\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.\]Equivalently, for real \(z\)
\[\operatorname{Ai}(z) := \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.\]Examples
Create an Airy function object:
>>> from sympy import airyai >>> from sympy.abc import z
>>> airyai(z) airyai(z)
Several special values are known:
>>> airyai(0) 3**(1/3)/(3*gamma(2/3)) >>> from sympy import oo >>> airyai(oo) 0 >>> airyai(-oo) 0
The Airy function obeys the mirror symmetry:
>>> from sympy import conjugate >>> conjugate(airyai(z)) airyai(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(airyai(z), z) airyaiprime(z) >>> diff(airyai(z), z, 2) z*airyai(z)
Series expansion is also supported:
>>> from sympy import series >>> series(airyai(z), z, 0, 3) 3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3)
We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:
>>> airyai(-2).evalf(50) 0.22740742820168557599192443603787379946077222541710
Rewrite \(\operatorname{Ai}(z)\) in terms of hypergeometric functions:
>>> from sympy import hyper >>> airyai(z).rewrite(hyper) -3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))
See also
airybiAiry function of the second kind.
airyaiprimeDerivative of the Airy function of the first kind.
airybiprimeDerivative of the Airy function of the second kind.
References
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class
sympy.functions.special.bessel.airybi(**kwargs)[source]¶ The Airy function \(\operatorname{Bi}\) of the second kind.
Explanation
The Airy function \(\operatorname{Bi}(z)\) is defined to be the function satisfying Airy’s differential equation
\[\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.\]Equivalently, for real \(z\)
\[\operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.\]Examples
Create an Airy function object:
>>> from sympy import airybi >>> from sympy.abc import z
>>> airybi(z) airybi(z)
Several special values are known:
>>> airybi(0) 3**(5/6)/(3*gamma(2/3)) >>> from sympy import oo >>> airybi(oo) oo >>> airybi(-oo) 0
The Airy function obeys the mirror symmetry:
>>> from sympy import conjugate >>> conjugate(airybi(z)) airybi(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(airybi(z), z) airybiprime(z) >>> diff(airybi(z), z, 2) z*airybi(z)
Series expansion is also supported:
>>> from sympy import series >>> series(airybi(z), z, 0, 3) 3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3)
We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:
>>> airybi(-2).evalf(50) -0.41230258795639848808323405461146104203453483447240
Rewrite \(\operatorname{Bi}(z)\) in terms of hypergeometric functions:
>>> from sympy import hyper >>> airybi(z).rewrite(hyper) 3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))
See also
airyaiAiry function of the first kind.
airyaiprimeDerivative of the Airy function of the first kind.
airybiprimeDerivative of the Airy function of the second kind.
References
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class
sympy.functions.special.bessel.airyaiprime(**kwargs)[source]¶ The derivative \(\operatorname{Ai}^\prime\) of the Airy function of the first kind.
Explanation
The Airy function \(\operatorname{Ai}^\prime(z)\) is defined to be the function
\[\operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}.\]Examples
Create an Airy function object:
>>> from sympy import airyaiprime >>> from sympy.abc import z
>>> airyaiprime(z) airyaiprime(z)
Several special values are known:
>>> airyaiprime(0) -3**(2/3)/(3*gamma(1/3)) >>> from sympy import oo >>> airyaiprime(oo) 0
The Airy function obeys the mirror symmetry:
>>> from sympy import conjugate >>> conjugate(airyaiprime(z)) airyaiprime(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(airyaiprime(z), z) z*airyai(z) >>> diff(airyaiprime(z), z, 2) z*airyaiprime(z) + airyai(z)
Series expansion is also supported:
>>> from sympy import series >>> series(airyaiprime(z), z, 0, 3) -3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3)
We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:
>>> airyaiprime(-2).evalf(50) 0.61825902074169104140626429133247528291577794512415
Rewrite \(\operatorname{Ai}^\prime(z)\) in terms of hypergeometric functions:
>>> from sympy import hyper >>> airyaiprime(z).rewrite(hyper) 3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3))
See also
airyaiAiry function of the first kind.
airybiAiry function of the second kind.
airybiprimeDerivative of the Airy function of the second kind.
References
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class
sympy.functions.special.bessel.airybiprime(**kwargs)[source]¶ The derivative \(\operatorname{Bi}^\prime\) of the Airy function of the first kind.
Explanation
The Airy function \(\operatorname{Bi}^\prime(z)\) is defined to be the function
\[\operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}.\]Examples
Create an Airy function object:
>>> from sympy import airybiprime >>> from sympy.abc import z
>>> airybiprime(z) airybiprime(z)
Several special values are known:
>>> airybiprime(0) 3**(1/6)/gamma(1/3) >>> from sympy import oo >>> airybiprime(oo) oo >>> airybiprime(-oo) 0
The Airy function obeys the mirror symmetry:
>>> from sympy import conjugate >>> conjugate(airybiprime(z)) airybiprime(conjugate(z))
Differentiation with respect to \(z\) is supported:
>>> from sympy import diff >>> diff(airybiprime(z), z) z*airybi(z) >>> diff(airybiprime(z), z, 2) z*airybiprime(z) + airybi(z)
Series expansion is also supported:
>>> from sympy import series >>> series(airybiprime(z), z, 0, 3) 3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3)
We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:
>>> airybiprime(-2).evalf(50) 0.27879516692116952268509756941098324140300059345163
Rewrite \(\operatorname{Bi}^\prime(z)\) in terms of hypergeometric functions:
>>> from sympy import hyper >>> airybiprime(z).rewrite(hyper) 3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3)
See also
airyaiAiry function of the first kind.
airybiAiry function of the second kind.
airyaiprimeDerivative of the Airy function of the first kind.
References
B-Splines¶
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sympy.functions.special.bsplines.bspline_basis(d, knots, n, x)[source]¶ The \(n\)-th B-spline at \(x\) of degree \(d\) with knots.
- Parameters
d : integer
degree of bspline
knots : list of integer values
list of knots points of bspline
n : integer
\(n\)-th B-spline
x : symbol
Explanation
B-Splines are piecewise polynomials of degree \(d\). They are defined on a set of knots, which is a sequence of integers or floats.
Examples
The 0th degree splines have a value of 1 on a single interval:
>>> from sympy import bspline_basis >>> from sympy.abc import x >>> d = 0 >>> knots = tuple(range(5)) >>> bspline_basis(d, knots, 0, x) Piecewise((1, (x >= 0) & (x <= 1)), (0, True))
For a given
(d, knots)there arelen(knots)-d-1B-splines defined, that are indexed byn(starting at 0).Here is an example of a cubic B-spline:
>>> bspline_basis(3, tuple(range(5)), 0, x) Piecewise((x**3/6, (x >= 0) & (x <= 1)), (-x**3/2 + 2*x**2 - 2*x + 2/3, (x >= 1) & (x <= 2)), (x**3/2 - 4*x**2 + 10*x - 22/3, (x >= 2) & (x <= 3)), (-x**3/6 + 2*x**2 - 8*x + 32/3, (x >= 3) & (x <= 4)), (0, True))
By repeating knot points, you can introduce discontinuities in the B-splines and their derivatives:
>>> d = 1 >>> knots = (0, 0, 2, 3, 4) >>> bspline_basis(d, knots, 0, x) Piecewise((1 - x/2, (x >= 0) & (x <= 2)), (0, True))
It is quite time consuming to construct and evaluate B-splines. If you need to evaluate a B-spline many times, it is best to lambdify them first:
>>> from sympy import lambdify >>> d = 3 >>> knots = tuple(range(10)) >>> b0 = bspline_basis(d, knots, 0, x) >>> f = lambdify(x, b0) >>> y = f(0.5)
See also
References
-
sympy.functions.special.bsplines.bspline_basis_set(d, knots, x)[source]¶ Return the
len(knots)-d-1B-splines at x of degree d with knots.- Parameters
d : integer
degree of bspline
knots : list of integers
list of knots points of bspline
x : symbol
Explanation
This function returns a list of piecewise polynomials that are the
len(knots)-d-1B-splines of degree d for the given knots. This function callsbspline_basis(d, knots, n, x)for different values of n.Examples
>>> from sympy import bspline_basis_set >>> from sympy.abc import x >>> d = 2 >>> knots = range(5) >>> splines = bspline_basis_set(d, knots, x) >>> splines [Piecewise((x**2/2, (x >= 0) & (x <= 1)), (-x**2 + 3*x - 3/2, (x >= 1) & (x <= 2)), (x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)), (0, True)), Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x <= 2)), (-x**2 + 5*x - 11/2, (x >= 2) & (x <= 3)), (x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)), (0, True))]
See also
-
sympy.functions.special.bsplines.interpolating_spline(d, x, X, Y)[source]¶ Return spline of degree d, passing through the given X and Y values.
- Parameters
d : integer
Degree of Bspline strictly greater than equal to one
x : symbol
X : list of strictly increasing integer values
list of X coordinates through which the spline passes
Y : list of strictly increasing integer values
list of Y coordinates through which the spline passes
Explanation
This function returns a piecewise function such that each part is a polynomial of degree not greater than d. The value of d must be 1 or greater and the values of X must be strictly increasing.
Examples
>>> from sympy import interpolating_spline >>> from sympy.abc import x >>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7]) Piecewise((3*x, (x >= 1) & (x <= 2)), (7 - x/2, (x >= 2) & (x <= 4)), (2*x/3 + 7/3, (x >= 4) & (x <= 7))) >>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3]) Piecewise((7*x**3/117 + 7*x**2/117 - 131*x/117 + 2, (x >= -2) & (x <= 1)), (10*x**3/117 - 2*x**2/117 - 122*x/117 + 77/39, (x >= 1) & (x <= 4)))
See also
Riemann Zeta and Related Functions¶
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class
sympy.functions.special.zeta_functions.zeta(**kwargs)[source]¶ Hurwitz zeta function (or Riemann zeta function).
Explanation
For \(\operatorname{Re}(a) > 0\) and \(\operatorname{Re}(s) > 1\), this function is defined as
\[\zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},\]where the standard choice of argument for \(n + a\) is used. For fixed \(a\) with \(\operatorname{Re}(a) > 0\) the Hurwitz zeta function admits a meromorphic continuation to all of \(\mathbb{C}\), it is an unbranched function with a simple pole at \(s = 1\).
Analytic continuation to other \(a\) is possible under some circumstances, but this is not typically done.
The Hurwitz zeta function is a special case of the Lerch transcendent:
\[\zeta(s, a) = \Phi(1, s, a).\]This formula defines an analytic continuation for all possible values of \(s\) and \(a\) (also \(\operatorname{Re}(a) < 0\)), see the documentation of
lerchphifor a description of the branching behavior.If no value is passed for \(a\), by this function assumes a default value of \(a = 1\), yielding the Riemann zeta function.
Examples
For \(a = 1\) the Hurwitz zeta function reduces to the famous Riemann zeta function:
\[\zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.\]>>> from sympy import zeta >>> from sympy.abc import s >>> zeta(s, 1) zeta(s) >>> zeta(s) zeta(s)
The Riemann zeta function can also be expressed using the Dirichlet eta function:
>>> from sympy import dirichlet_eta >>> zeta(s).rewrite(dirichlet_eta) dirichlet_eta(s)/(1 - 2**(1 - s))
The Riemann zeta function at positive even integer and negative odd integer values is related to the Bernoulli numbers:
>>> zeta(2) pi**2/6 >>> zeta(4) pi**4/90 >>> zeta(-1) -1/12
The specific formulae are:
\[\zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}\]\[\zeta(-n) = -\frac{B_{n+1}}{n+1}\]At negative even integers the Riemann zeta function is zero:
>>> zeta(-4) 0
No closed-form expressions are known at positive odd integers, but numerical evaluation is possible:
>>> zeta(3).n() 1.20205690315959
The derivative of \(\zeta(s, a)\) with respect to \(a\) can be computed:
>>> from sympy.abc import a >>> zeta(s, a).diff(a) -s*zeta(s + 1, a)
However the derivative with respect to \(s\) has no useful closed form expression:
>>> zeta(s, a).diff(s) Derivative(zeta(s, a), s)
The Hurwitz zeta function can be expressed in terms of the Lerch transcendent,
lerchphi:>>> from sympy import lerchphi >>> zeta(s, a).rewrite(lerchphi) lerchphi(1, s, a)
See also
References
-
class
sympy.functions.special.zeta_functions.dirichlet_eta(**kwargs)[source]¶ Dirichlet eta function.
Explanation
For \(\operatorname{Re}(s) > 0\), this function is defined as
\[\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}.\]It admits a unique analytic continuation to all of \(\mathbb{C}\). It is an entire, unbranched function.
Examples
The Dirichlet eta function is closely related to the Riemann zeta function:
>>> from sympy import dirichlet_eta, zeta >>> from sympy.abc import s >>> dirichlet_eta(s).rewrite(zeta) (1 - 2**(1 - s))*zeta(s)
See also
References
-
class
sympy.functions.special.zeta_functions.polylog(**kwargs)[source]¶ Polylogarithm function.
Explanation
For \(|z| < 1\) and \(s \in \mathbb{C}\), the polylogarithm is defined by
\[\operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},\]where the standard branch of the argument is used for \(n\). It admits an analytic continuation which is branched at \(z=1\) (notably not on the sheet of initial definition), \(z=0\) and \(z=\infty\).
The name polylogarithm comes from the fact that for \(s=1\), the polylogarithm is related to the ordinary logarithm (see examples), and that
\[\operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.\]The polylogarithm is a special case of the Lerch transcendent:
\[\operatorname{Li}_{s}(z) = z \Phi(z, s, 1).\]Examples
For \(z \in \{0, 1, -1\}\), the polylogarithm is automatically expressed using other functions:
>>> from sympy import polylog >>> from sympy.abc import s >>> polylog(s, 0) 0 >>> polylog(s, 1) zeta(s) >>> polylog(s, -1) -dirichlet_eta(s)
If \(s\) is a negative integer, \(0\) or \(1\), the polylogarithm can be expressed using elementary functions. This can be done using
expand_func():>>> from sympy import expand_func >>> from sympy.abc import z >>> expand_func(polylog(1, z)) -log(1 - z) >>> expand_func(polylog(0, z)) z/(1 - z)
The derivative with respect to \(z\) can be computed in closed form:
>>> polylog(s, z).diff(z) polylog(s - 1, z)/z
The polylogarithm can be expressed in terms of the lerch transcendent:
>>> from sympy import lerchphi >>> polylog(s, z).rewrite(lerchphi) z*lerchphi(z, s, 1)
-
class
sympy.functions.special.zeta_functions.lerchphi(**kwargs)[source]¶ Lerch transcendent (Lerch phi function).
Explanation
For \(\operatorname{Re}(a) > 0\), \(|z| < 1\) and \(s \in \mathbb{C}\), the Lerch transcendent is defined as
\[\Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},\]where the standard branch of the argument is used for \(n + a\), and by analytic continuation for other values of the parameters.
A commonly used related function is the Lerch zeta function, defined by
\[L(q, s, a) = \Phi(e^{2\pi i q}, s, a).\]Analytic Continuation and Branching Behavior
It can be shown that
\[\Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.\]This provides the analytic continuation to \(\operatorname{Re}(a) \le 0\).
Assume now \(\operatorname{Re}(a) > 0\). The integral representation
\[\Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}} \frac{\mathrm{d}t}{\Gamma(s)}\]provides an analytic continuation to \(\mathbb{C} - [1, \infty)\). Finally, for \(x \in (1, \infty)\) we find
\[\lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a) -\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a) = \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},\]using the standard branch for both \(\log{x}\) and \(\log{\log{x}}\) (a branch of \(\log{\log{x}}\) is needed to evaluate \(\log{x}^{s-1}\)). This concludes the analytic continuation. The Lerch transcendent is thus branched at \(z \in \{0, 1, \infty\}\) and \(a \in \mathbb{Z}_{\le 0}\). For fixed \(z, a\) outside these branch points, it is an entire function of \(s\).
Examples
The Lerch transcendent is a fairly general function, for this reason it does not automatically evaluate to simpler functions. Use
expand_func()to achieve this.If \(z=1\), the Lerch transcendent reduces to the Hurwitz zeta function:
>>> from sympy import lerchphi, expand_func >>> from sympy.abc import z, s, a >>> expand_func(lerchphi(1, s, a)) zeta(s, a)
More generally, if \(z\) is a root of unity, the Lerch transcendent reduces to a sum of Hurwitz zeta functions:
>>> expand_func(lerchphi(-1, s, a)) 2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, a/2 + 1/2)
If \(a=1\), the Lerch transcendent reduces to the polylogarithm:
>>> expand_func(lerchphi(z, s, 1)) polylog(s, z)/z
More generally, if \(a\) is rational, the Lerch transcendent reduces to a sum of polylogarithms:
>>> from sympy import S >>> expand_func(lerchphi(z, s, S(1)/2)) 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) - polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z)) >>> expand_func(lerchphi(z, s, S(3)/2)) -2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) - polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z
The derivatives with respect to \(z\) and \(a\) can be computed in closed form:
>>> lerchphi(z, s, a).diff(z) (-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z >>> lerchphi(z, s, a).diff(a) -s*lerchphi(z, s + 1, a)
References
-
class
sympy.functions.special.zeta_functions.stieltjes(**kwargs)[source]¶ Represents Stieltjes constants, \(\gamma_{k}\) that occur in Laurent Series expansion of the Riemann zeta function.
Examples
>>> from sympy import stieltjes >>> from sympy.abc import n, m >>> stieltjes(n) stieltjes(n)
The zero’th stieltjes constant:
>>> stieltjes(0) EulerGamma >>> stieltjes(0, 1) EulerGamma
For generalized stieltjes constants:
>>> stieltjes(n, m) stieltjes(n, m)
Constants are only defined for integers >= 0:
>>> stieltjes(-1) zoo
References
Hypergeometric Functions¶
-
class
sympy.functions.special.hyper.hyper(ap, bq, z, **kwargs)[source]¶ The generalized hypergeometric function is defined by a series where the ratios of successive terms are a rational function of the summation index. When convergent, it is continued analytically to the largest possible domain.
Explanation
The hypergeometric function depends on two vectors of parameters, called the numerator parameters \(a_p\), and the denominator parameters \(b_q\). It also has an argument \(z\). The series definition is
\[\begin{split}{}_pF_q\left(\begin{matrix} a_1, \cdots, a_p \\ b_1, \cdots, b_q \end{matrix} \middle| z \right) = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!},\end{split}\]where \((a)_n = (a)(a+1)\cdots(a+n-1)\) denotes the rising factorial.
If one of the \(b_q\) is a non-positive integer then the series is undefined unless one of the \(a_p\) is a larger (i.e., smaller in magnitude) non-positive integer. If none of the \(b_q\) is a non-positive integer and one of the \(a_p\) is a non-positive integer, then the series reduces to a polynomial. To simplify the following discussion, we assume that none of the \(a_p\) or \(b_q\) is a non-positive integer. For more details, see the references.
The series converges for all \(z\) if \(p \le q\), and thus defines an entire single-valued function in this case. If \(p = q+1\) the series converges for \(|z| < 1\), and can be continued analytically into a half-plane. If \(p > q+1\) the series is divergent for all \(z\).
Please note the hypergeometric function constructor currently does not check if the parameters actually yield a well-defined function.
Examples
The parameters \(a_p\) and \(b_q\) can be passed as arbitrary iterables, for example:
>>> from sympy.functions import hyper >>> from sympy.abc import x, n, a >>> hyper((1, 2, 3), [3, 4], x) hyper((1, 2, 3), (3, 4), x)
There is also pretty printing (it looks better using Unicode):
>>> from sympy import pprint >>> pprint(hyper((1, 2, 3), [3, 4], x), use_unicode=False) _ |_ /1, 2, 3 | \ | | | x| 3 2 \ 3, 4 | /
The parameters must always be iterables, even if they are vectors of length one or zero:
>>> hyper((1, ), [], x) hyper((1,), (), x)
But of course they may be variables (but if they depend on \(x\) then you should not expect much implemented functionality):
>>> hyper((n, a), (n**2,), x) hyper((n, a), (n**2,), x)
The hypergeometric function generalizes many named special functions. The function
hyperexpand()tries to express a hypergeometric function using named special functions. For example:>>> from sympy import hyperexpand >>> hyperexpand(hyper([], [], x)) exp(x)
You can also use
expand_func():>>> from sympy import expand_func >>> expand_func(x*hyper([1, 1], [2], -x)) log(x + 1)
More examples:
>>> from sympy import S >>> hyperexpand(hyper([], [S(1)/2], -x**2/4)) cos(x) >>> hyperexpand(x*hyper([S(1)/2, S(1)/2], [S(3)/2], x**2)) asin(x)
We can also sometimes
hyperexpand()parametric functions:>>> from sympy.abc import a >>> hyperexpand(hyper([-a], [], x)) (1 - x)**a
See also
References
- R402
Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1
- R403
https://en.wikipedia.org/wiki/Generalized_hypergeometric_function
-
property
ap¶ Numerator parameters of the hypergeometric function.
-
property
argument¶ Argument of the hypergeometric function.
-
property
bq¶ Denominator parameters of the hypergeometric function.
-
property
convergence_statement¶ Return a condition on z under which the series converges.
-
property
eta¶ A quantity related to the convergence of the series.
-
property
radius_of_convergence¶ Compute the radius of convergence of the defining series.
Explanation
Note that even if this is not
oo, the function may still be evaluated outside of the radius of convergence by analytic continuation. But if this is zero, then the function is not actually defined anywhere else.Examples
>>> from sympy.functions import hyper >>> from sympy.abc import z >>> hyper((1, 2), [3], z).radius_of_convergence 1 >>> hyper((1, 2, 3), [4], z).radius_of_convergence 0 >>> hyper((1, 2), (3, 4), z).radius_of_convergence oo
-
class
sympy.functions.special.hyper.meijerg(*args, **kwargs)[source]¶ The Meijer G-function is defined by a Mellin-Barnes type integral that resembles an inverse Mellin transform. It generalizes the hypergeometric functions.
Explanation
The Meijer G-function depends on four sets of parameters. There are “numerator parameters” \(a_1, \ldots, a_n\) and \(a_{n+1}, \ldots, a_p\), and there are “denominator parameters” \(b_1, \ldots, b_m\) and \(b_{m+1}, \ldots, b_q\). Confusingly, it is traditionally denoted as follows (note the position of \(m\), \(n\), \(p\), \(q\), and how they relate to the lengths of the four parameter vectors):
\[\begin{split}G_{p,q}^{m,n} \left(\begin{matrix}a_1, \cdots, a_n & a_{n+1}, \cdots, a_p \\ b_1, \cdots, b_m & b_{m+1}, \cdots, b_q \end{matrix} \middle| z \right).\end{split}\]However, in SymPy the four parameter vectors are always available separately (see examples), so that there is no need to keep track of the decorating sub- and super-scripts on the G symbol.
The G function is defined as the following integral:
\[\frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s) \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s) \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,\]where \(\Gamma(z)\) is the gamma function. There are three possible contours which we will not describe in detail here (see the references). If the integral converges along more than one of them, the definitions agree. The contours all separate the poles of \(\Gamma(1-a_j+s)\) from the poles of \(\Gamma(b_k-s)\), so in particular the G function is undefined if \(a_j - b_k \in \mathbb{Z}_{>0}\) for some \(j \le n\) and \(k \le m\).
The conditions under which one of the contours yields a convergent integral are complicated and we do not state them here, see the references.
Please note currently the Meijer G-function constructor does not check any convergence conditions.
Examples
You can pass the parameters either as four separate vectors:
>>> from sympy.functions import meijerg >>> from sympy.abc import x, a >>> from sympy.core.containers import Tuple >>> from sympy import pprint >>> pprint(meijerg((1, 2), (a, 4), (5,), [], x), use_unicode=False) __1, 2 /1, 2 a, 4 | \ /__ | | x| \_|4, 1 \ 5 | /
Or as two nested vectors:
>>> pprint(meijerg([(1, 2), (3, 4)], ([5], Tuple()), x), use_unicode=False) __1, 2 /1, 2 3, 4 | \ /__ | | x| \_|4, 1 \ 5 | /
As with the hypergeometric function, the parameters may be passed as arbitrary iterables. Vectors of length zero and one also have to be passed as iterables. The parameters need not be constants, but if they depend on the argument then not much implemented functionality should be expected.
All the subvectors of parameters are available:
>>> from sympy import pprint >>> g = meijerg([1], [2], [3], [4], x) >>> pprint(g, use_unicode=False) __1, 1 /1 2 | \ /__ | | x| \_|2, 2 \3 4 | / >>> g.an (1,) >>> g.ap (1, 2) >>> g.aother (2,) >>> g.bm (3,) >>> g.bq (3, 4) >>> g.bother (4,)
The Meijer G-function generalizes the hypergeometric functions. In some cases it can be expressed in terms of hypergeometric functions, using Slater’s theorem. For example:
>>> from sympy import hyperexpand >>> from sympy.abc import a, b, c >>> hyperexpand(meijerg([a], [], [c], [b], x), allow_hyper=True) x**c*gamma(-a + c + 1)*hyper((-a + c + 1,), (-b + c + 1,), -x)/gamma(-b + c + 1)
Thus the Meijer G-function also subsumes many named functions as special cases. You can use
expand_func()orhyperexpand()to (try to) rewrite a Meijer G-function in terms of named special functions. For example:>>> from sympy import expand_func, S >>> expand_func(meijerg([[],[]], [[0],[]], -x)) exp(x) >>> hyperexpand(meijerg([[],[]], [[S(1)/2],[0]], (x/2)**2)) sin(x)/sqrt(pi)
See also
References
-
property
an¶ First set of numerator parameters.
-
property
aother¶ Second set of numerator parameters.
-
property
ap¶ Combined numerator parameters.
-
property
argument¶ Argument of the Meijer G-function.
-
property
bm¶ First set of denominator parameters.
-
property
bother¶ Second set of denominator parameters.
-
property
bq¶ Combined denominator parameters.
-
property
delta¶ A quantity related to the convergence region of the integral, c.f. references.
-
get_period()[source]¶ Return a number \(P\) such that \(G(x*exp(I*P)) == G(x)\).
Examples
>>> from sympy.functions.special.hyper import meijerg >>> from sympy.abc import z >>> from sympy import pi, S
>>> meijerg([1], [], [], [], z).get_period() 2*pi >>> meijerg([pi], [], [], [], z).get_period() oo >>> meijerg([1, 2], [], [], [], z).get_period() oo >>> meijerg([1,1], [2], [1, S(1)/2, S(1)/3], [1], z).get_period() 12*pi
-
property
is_number¶ Returns true if expression has numeric data only.
-
property
nu¶ A quantity related to the convergence region of the integral, c.f. references.
-
property
-
class
sympy.functions.special.hyper.appellf1(**kwargs)[source]¶ This is the Appell hypergeometric function of two variables as:
\[F_1(a,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}} \frac{x^m y^n}{m! n!}.\]Examples
>>> from sympy.functions.special.hyper import appellf1 >>> from sympy import symbols >>> x, y, a, b1, b2, c = symbols('x y a b1 b2 c') >>> appellf1(2., 1., 6., 4., 5., 6.) 0.0063339426292673 >>> appellf1(12., 12., 6., 4., 0.5, 0.12) 172870711.659936 >>> appellf1(40, 2, 6, 4, 15, 60) appellf1(40, 2, 6, 4, 15, 60) >>> appellf1(20., 12., 10., 3., 0.5, 0.12) 15605338197184.4 >>> appellf1(40, 2, 6, 4, x, y) appellf1(40, 2, 6, 4, x, y) >>> appellf1(a, b1, b2, c, x, y) appellf1(a, b1, b2, c, x, y)
References
Elliptic integrals¶
-
class
sympy.functions.special.elliptic_integrals.elliptic_k(**kwargs)[source]¶ The complete elliptic integral of the first kind, defined by
\[K(m) = F\left(\tfrac{\pi}{2}\middle| m\right)\]where \(F\left(z\middle| m\right)\) is the Legendre incomplete elliptic integral of the first kind.
Explanation
The function \(K(m)\) is a single-valued function on the complex plane with branch cut along the interval \((1, \infty)\).
Note that our notation defines the incomplete elliptic integral in terms of the parameter \(m\) instead of the elliptic modulus (eccentricity) \(k\). In this case, the parameter \(m\) is defined as \(m=k^2\).
Examples
>>> from sympy import elliptic_k, I >>> from sympy.abc import m >>> elliptic_k(0) pi/2 >>> elliptic_k(1.0 + I) 1.50923695405127 + 0.625146415202697*I >>> elliptic_k(m).series(n=3) pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3)
See also
References
-
class
sympy.functions.special.elliptic_integrals.elliptic_f(**kwargs)[source]¶ The Legendre incomplete elliptic integral of the first kind, defined by
\[F\left(z\middle| m\right) = \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}\]Explanation
This function reduces to a complete elliptic integral of the first kind, \(K(m)\), when \(z = \pi/2\).
Note that our notation defines the incomplete elliptic integral in terms of the parameter \(m\) instead of the elliptic modulus (eccentricity) \(k\). In this case, the parameter \(m\) is defined as \(m=k^2\).
Examples
>>> from sympy import elliptic_f, I >>> from sympy.abc import z, m >>> elliptic_f(z, m).series(z) z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6) >>> elliptic_f(3.0 + I/2, 1.0 + I) 2.909449841483 + 1.74720545502474*I
See also
References
-
class
sympy.functions.special.elliptic_integrals.elliptic_e(**kwargs)[source]¶ Called with two arguments \(z\) and \(m\), evaluates the incomplete elliptic integral of the second kind, defined by
\[E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt\]Called with a single argument \(m\), evaluates the Legendre complete elliptic integral of the second kind
\[E(m) = E\left(\tfrac{\pi}{2}\middle| m\right)\]Explanation
The function \(E(m)\) is a single-valued function on the complex plane with branch cut along the interval \((1, \infty)\).
Note that our notation defines the incomplete elliptic integral in terms of the parameter \(m\) instead of the elliptic modulus (eccentricity) \(k\). In this case, the parameter \(m\) is defined as \(m=k^2\).
Examples
>>> from sympy import elliptic_e, I >>> from sympy.abc import z, m >>> elliptic_e(z, m).series(z) z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6) >>> elliptic_e(m).series(n=4) pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4) >>> elliptic_e(1 + I, 2 - I/2).n() 1.55203744279187 + 0.290764986058437*I >>> elliptic_e(0) pi/2 >>> elliptic_e(2.0 - I) 0.991052601328069 + 0.81879421395609*I
References
-
class
sympy.functions.special.elliptic_integrals.elliptic_pi(**kwargs)[source]¶ Called with three arguments \(n\), \(z\) and \(m\), evaluates the Legendre incomplete elliptic integral of the third kind, defined by
\[\Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt} {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}\]Called with two arguments \(n\) and \(m\), evaluates the complete elliptic integral of the third kind:
\[\Pi\left(n\middle| m\right) = \Pi\left(n; \tfrac{\pi}{2}\middle| m\right)\]Explanation
Note that our notation defines the incomplete elliptic integral in terms of the parameter \(m\) instead of the elliptic modulus (eccentricity) \(k\). In this case, the parameter \(m\) is defined as \(m=k^2\).
Examples
>>> from sympy import elliptic_pi, I >>> from sympy.abc import z, n, m >>> elliptic_pi(n, z, m).series(z, n=4) z + z**3*(m/6 + n/3) + O(z**4) >>> elliptic_pi(0.5 + I, 1.0 - I, 1.2) 2.50232379629182 - 0.760939574180767*I >>> elliptic_pi(0, 0) pi/2 >>> elliptic_pi(1.0 - I/3, 2.0 + I) 3.29136443417283 + 0.32555634906645*I
References
Mathieu Functions¶
-
class
sympy.functions.special.mathieu_functions.MathieuBase(**kwargs)[source]¶ Abstract base class for Mathieu functions.
This class is meant to reduce code duplication.
-
class
sympy.functions.special.mathieu_functions.mathieus(**kwargs)[source]¶ The Mathieu Sine function \(S(a,q,z)\).
Explanation
This function is one solution of the Mathieu differential equation:
\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]The other solution is the Mathieu Cosine function.
Examples
>>> from sympy import diff, mathieus >>> from sympy.abc import a, q, z
>>> mathieus(a, q, z) mathieus(a, q, z)
>>> mathieus(a, 0, z) sin(sqrt(a)*z)
>>> diff(mathieus(a, q, z), z) mathieusprime(a, q, z)
See also
mathieucMathieu cosine function.
mathieusprimeDerivative of Mathieu sine function.
mathieucprimeDerivative of Mathieu cosine function.
References
-
class
sympy.functions.special.mathieu_functions.mathieuc(**kwargs)[source]¶ The Mathieu Cosine function \(C(a,q,z)\).
Explanation
This function is one solution of the Mathieu differential equation:
\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]The other solution is the Mathieu Sine function.
Examples
>>> from sympy import diff, mathieuc >>> from sympy.abc import a, q, z
>>> mathieuc(a, q, z) mathieuc(a, q, z)
>>> mathieuc(a, 0, z) cos(sqrt(a)*z)
>>> diff(mathieuc(a, q, z), z) mathieucprime(a, q, z)
See also
mathieusMathieu sine function
mathieusprimeDerivative of Mathieu sine function
mathieucprimeDerivative of Mathieu cosine function
References
-
class
sympy.functions.special.mathieu_functions.mathieusprime(**kwargs)[source]¶ The derivative \(S^{\prime}(a,q,z)\) of the Mathieu Sine function.
Explanation
This function is one solution of the Mathieu differential equation:
\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]The other solution is the Mathieu Cosine function.
Examples
>>> from sympy import diff, mathieusprime >>> from sympy.abc import a, q, z
>>> mathieusprime(a, q, z) mathieusprime(a, q, z)
>>> mathieusprime(a, 0, z) sqrt(a)*cos(sqrt(a)*z)
>>> diff(mathieusprime(a, q, z), z) (-a + 2*q*cos(2*z))*mathieus(a, q, z)
See also
mathieusMathieu sine function
mathieucMathieu cosine function
mathieucprimeDerivative of Mathieu cosine function
References
-
class
sympy.functions.special.mathieu_functions.mathieucprime(**kwargs)[source]¶ The derivative \(C^{\prime}(a,q,z)\) of the Mathieu Cosine function.
Explanation
This function is one solution of the Mathieu differential equation:
\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]The other solution is the Mathieu Sine function.
Examples
>>> from sympy import diff, mathieucprime >>> from sympy.abc import a, q, z
>>> mathieucprime(a, q, z) mathieucprime(a, q, z)
>>> mathieucprime(a, 0, z) -sqrt(a)*sin(sqrt(a)*z)
>>> diff(mathieucprime(a, q, z), z) (-a + 2*q*cos(2*z))*mathieuc(a, q, z)
See also
mathieusMathieu sine function
mathieucMathieu cosine function
mathieusprimeDerivative of Mathieu sine function
References
Orthogonal Polynomials¶
This module mainly implements special orthogonal polynomials.
See also functions.combinatorial.numbers which contains some combinatorial polynomials.
Jacobi Polynomials¶
-
class
sympy.functions.special.polynomials.jacobi(**kwargs)[source]¶ Jacobi polynomial \(P_n^{\left(\alpha, \beta\right)}(x)\).
Explanation
jacobi(n, alpha, beta, x)gives the nth Jacobi polynomial in x, \(P_n^{\left(\alpha, \beta\right)}(x)\).The Jacobi polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x\right)^\alpha \left(1+x\right)^\beta\).
Examples
>>> from sympy import jacobi, S, conjugate, diff >>> from sympy.abc import a, b, n, x
>>> jacobi(0, a, b, x) 1 >>> jacobi(1, a, b, x) a/2 - b/2 + x*(a/2 + b/2 + 1) >>> jacobi(2, a, b, x) a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2
>>> jacobi(n, a, b, x) jacobi(n, a, b, x)
>>> jacobi(n, a, a, x) RisingFactorial(a + 1, n)*gegenbauer(n, a + 1/2, x)/RisingFactorial(2*a + 1, n)
>>> jacobi(n, 0, 0, x) legendre(n, x)
>>> jacobi(n, S(1)/2, S(1)/2, x) RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)
>>> jacobi(n, -S(1)/2, -S(1)/2, x) RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)
>>> jacobi(n, a, b, -x) (-1)**n*jacobi(n, b, a, x)
>>> jacobi(n, a, b, 0) 2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1)) >>> jacobi(n, a, b, 1) RisingFactorial(a + 1, n)/factorial(n)
>>> conjugate(jacobi(n, a, b, x)) jacobi(n, conjugate(a), conjugate(b), conjugate(x))
>>> diff(jacobi(n,a,b,x), x) (a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)
See also
gegenbauer,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
-
sympy.functions.special.polynomials.jacobi_normalized(n, a, b, x)[source]¶ Jacobi polynomial \(P_n^{\left(\alpha, \beta\right)}(x)\).
- Parameters
n : integer degree of polynomial
a : alpha value
b : beta value
x : symbol
Explanation
jacobi_normalized(n, alpha, beta, x)gives the nth Jacobi polynomial in x, \(P_n^{\left(\alpha, \beta\right)}(x)\).The Jacobi polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x\right)^\alpha \left(1+x\right)^\beta\).
This functions returns the polynomials normilzed:
\[\int_{-1}^{1} P_m^{\left(\alpha, \beta\right)}(x) P_n^{\left(\alpha, \beta\right)}(x) (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x = \delta_{m,n}\]Examples
>>> from sympy import jacobi_normalized >>> from sympy.abc import n,a,b,x
>>> jacobi_normalized(n, a, b, x) jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1)))
See also
gegenbauer,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
Gegenbauer Polynomials¶
-
class
sympy.functions.special.polynomials.gegenbauer(**kwargs)[source]¶ Gegenbauer polynomial \(C_n^{\left(\alpha\right)}(x)\).
Explanation
gegenbauer(n, alpha, x)gives the nth Gegenbauer polynomial in x, \(C_n^{\left(\alpha\right)}(x)\).The Gegenbauer polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x^2\right)^{\alpha-\frac{1}{2}}\).
Examples
>>> from sympy import gegenbauer, conjugate, diff >>> from sympy.abc import n,a,x >>> gegenbauer(0, a, x) 1 >>> gegenbauer(1, a, x) 2*a*x >>> gegenbauer(2, a, x) -a + x**2*(2*a**2 + 2*a) >>> gegenbauer(3, a, x) x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
>>> gegenbauer(n, a, x) gegenbauer(n, a, x) >>> gegenbauer(n, a, -x) (-1)**n*gegenbauer(n, a, x)
>>> gegenbauer(n, a, 0) 2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1)) >>> gegenbauer(n, a, 1) gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))
>>> conjugate(gegenbauer(n, a, x)) gegenbauer(n, conjugate(a), conjugate(x))
>>> diff(gegenbauer(n, a, x), x) 2*a*gegenbauer(n - 1, a + 1, x)
See also
jacobi,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
Chebyshev Polynomials¶
-
class
sympy.functions.special.polynomials.chebyshevt(**kwargs)[source]¶ Chebyshev polynomial of the first kind, \(T_n(x)\).
Explanation
chebyshevt(n, x)gives the nth Chebyshev polynomial (of the first kind) in x, \(T_n(x)\).The Chebyshev polynomials of the first kind are orthogonal on \([-1, 1]\) with respect to the weight \(\frac{1}{\sqrt{1-x^2}}\).
Examples
>>> from sympy import chebyshevt, diff >>> from sympy.abc import n,x >>> chebyshevt(0, x) 1 >>> chebyshevt(1, x) x >>> chebyshevt(2, x) 2*x**2 - 1
>>> chebyshevt(n, x) chebyshevt(n, x) >>> chebyshevt(n, -x) (-1)**n*chebyshevt(n, x) >>> chebyshevt(-n, x) chebyshevt(n, x)
>>> chebyshevt(n, 0) cos(pi*n/2) >>> chebyshevt(n, -1) (-1)**n
>>> diff(chebyshevt(n, x), x) n*chebyshevu(n - 1, x)
See also
jacobi,gegenbauer,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
-
class
sympy.functions.special.polynomials.chebyshevu(**kwargs)[source]¶ Chebyshev polynomial of the second kind, \(U_n(x)\).
Explanation
chebyshevu(n, x)gives the nth Chebyshev polynomial of the second kind in x, \(U_n(x)\).The Chebyshev polynomials of the second kind are orthogonal on \([-1, 1]\) with respect to the weight \(\sqrt{1-x^2}\).
Examples
>>> from sympy import chebyshevu, diff >>> from sympy.abc import n,x >>> chebyshevu(0, x) 1 >>> chebyshevu(1, x) 2*x >>> chebyshevu(2, x) 4*x**2 - 1
>>> chebyshevu(n, x) chebyshevu(n, x) >>> chebyshevu(n, -x) (-1)**n*chebyshevu(n, x) >>> chebyshevu(-n, x) -chebyshevu(n - 2, x)
>>> chebyshevu(n, 0) cos(pi*n/2) >>> chebyshevu(n, 1) n + 1
>>> diff(chebyshevu(n, x), x) (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
-
class
sympy.functions.special.polynomials.chebyshevt_root(**kwargs)[source]¶ chebyshev_root(n, k)returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the first kind; that is, if 0 <= k < n,chebyshevt(n, chebyshevt_root(n, k)) == 0.Examples
>>> from sympy import chebyshevt, chebyshevt_root >>> chebyshevt_root(3, 2) -sqrt(3)/2 >>> chebyshevt(3, chebyshevt_root(3, 2)) 0
See also
jacobi,gegenbauer,chebyshevt,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_poly
-
class
sympy.functions.special.polynomials.chebyshevu_root(**kwargs)[source]¶ chebyshevu_root(n, k)returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the second kind; that is, if 0 <= k < n,chebyshevu(n, chebyshevu_root(n, k)) == 0.Examples
>>> from sympy import chebyshevu, chebyshevu_root >>> chebyshevu_root(3, 2) -sqrt(2)/2 >>> chebyshevu(3, chebyshevu_root(3, 2)) 0
See also
chebyshevt,chebyshevt_root,chebyshevu,legendre,assoc_legendre,hermite,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_poly
Legendre Polynomials¶
-
class
sympy.functions.special.polynomials.legendre(**kwargs)[source]¶ legendre(n, x)gives the nth Legendre polynomial of x, \(P_n(x)\)Explanation
The Legendre polynomials are orthogonal on [-1, 1] with respect to the constant weight 1. They satisfy \(P_n(1) = 1\) for all n; further, \(P_n\) is odd for odd n and even for even n.
Examples
>>> from sympy import legendre, diff >>> from sympy.abc import x, n >>> legendre(0, x) 1 >>> legendre(1, x) x >>> legendre(2, x) 3*x**2/2 - 1/2 >>> legendre(n, x) legendre(n, x) >>> diff(legendre(n,x), x) n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,assoc_legendre,hermite,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
-
class
sympy.functions.special.polynomials.assoc_legendre(**kwargs)[source]¶ assoc_legendre(n, m, x)gives \(P_n^m(x)\), where n and m are the degree and order or an expression which is related to the nth order Legendre polynomial, \(P_n(x)\) in the following manner:\[P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}} \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}\]Explanation
Associated Legendre polynomials are orthogonal on [-1, 1] with:
weight = 1 for the same m, and different n.
weight = 1/(1-x**2) for the same n, and different m.
Examples
>>> from sympy import assoc_legendre >>> from sympy.abc import x, m, n >>> assoc_legendre(0,0, x) 1 >>> assoc_legendre(1,0, x) x >>> assoc_legendre(1,1, x) -sqrt(1 - x**2) >>> assoc_legendre(n,m,x) assoc_legendre(n, m, x)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,hermite,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
Hermite Polynomials¶
-
class
sympy.functions.special.polynomials.hermite(**kwargs)[source]¶ hermite(n, x)gives the nth Hermite polynomial in x, \(H_n(x)\)Explanation
The Hermite polynomials are orthogonal on \((-\infty, \infty)\) with respect to the weight \(\exp\left(-x^2\right)\).
Examples
>>> from sympy import hermite, diff >>> from sympy.abc import x, n >>> hermite(0, x) 1 >>> hermite(1, x) 2*x >>> hermite(2, x) 4*x**2 - 2 >>> hermite(n, x) hermite(n, x) >>> diff(hermite(n,x), x) 2*n*hermite(n - 1, x) >>> hermite(n, -x) (-1)**n*hermite(n, x)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,laguerre,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
Laguerre Polynomials¶
-
class
sympy.functions.special.polynomials.laguerre(**kwargs)[source]¶ Returns the nth Laguerre polynomial in x, \(L_n(x)\).
- Parameters
n : int
Degree of Laguerre polynomial. Must be
n >= 0.
Examples
>>> from sympy import laguerre, diff >>> from sympy.abc import x, n >>> laguerre(0, x) 1 >>> laguerre(1, x) 1 - x >>> laguerre(2, x) x**2/2 - 2*x + 1 >>> laguerre(3, x) -x**3/6 + 3*x**2/2 - 3*x + 1
>>> laguerre(n, x) laguerre(n, x)
>>> diff(laguerre(n, x), x) -assoc_laguerre(n - 1, 1, x)
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,assoc_laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
-
class
sympy.functions.special.polynomials.assoc_laguerre(**kwargs)[source]¶ Returns the nth generalized Laguerre polynomial in x, \(L_n(x)\).
- Parameters
n : int
Degree of Laguerre polynomial. Must be
n >= 0.alpha : Expr
Arbitrary expression. For
alpha=0regular Laguerre polynomials will be generated.
Examples
>>> from sympy import assoc_laguerre, diff >>> from sympy.abc import x, n, a >>> assoc_laguerre(0, a, x) 1 >>> assoc_laguerre(1, a, x) a - x + 1 >>> assoc_laguerre(2, a, x) a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1 >>> assoc_laguerre(3, a, x) a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) + x*(-a**2/2 - 5*a/2 - 3) + 1
>>> assoc_laguerre(n, a, 0) binomial(a + n, a)
>>> assoc_laguerre(n, a, x) assoc_laguerre(n, a, x)
>>> assoc_laguerre(n, 0, x) laguerre(n, x)
>>> diff(assoc_laguerre(n, a, x), x) -assoc_laguerre(n - 1, a + 1, x)
>>> diff(assoc_laguerre(n, a, x), a) Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))
See also
jacobi,gegenbauer,chebyshevt,chebyshevt_root,chebyshevu,chebyshevu_root,legendre,assoc_legendre,hermite,laguerre,sympy.polys.orthopolys.jacobi_poly,sympy.polys.orthopolys.gegenbauer_poly,sympy.polys.orthopolys.chebyshevt_poly,sympy.polys.orthopolys.chebyshevu_poly,sympy.polys.orthopolys.hermite_poly,sympy.polys.orthopolys.legendre_poly,sympy.polys.orthopolys.laguerre_polyReferences
Spherical Harmonics¶
-
class
sympy.functions.special.spherical_harmonics.Ynm(**kwargs)[source]¶ Spherical harmonics defined as
\[Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}} \exp(i m \varphi) \mathrm{P}_n^m\left(\cos(\theta)\right)\]Explanation
Ynm()gives the spherical harmonic function of order \(n\) and \(m\) in \(\theta\) and \(\varphi\), \(Y_n^m(\theta, \varphi)\). The four parameters are as follows: \(n \geq 0\) an integer and \(m\) an integer such that \(-n \leq m \leq n\) holds. The two angles are real-valued with \(\theta \in [0, \pi]\) and \(\varphi \in [0, 2\pi]\).Examples
>>> from sympy import Ynm, Symbol, simplify >>> from sympy.abc import n,m >>> theta = Symbol("theta") >>> phi = Symbol("phi")
>>> Ynm(n, m, theta, phi) Ynm(n, m, theta, phi)
Several symmetries are known, for the order:
>>> Ynm(n, -m, theta, phi) (-1)**m*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
As well as for the angles:
>>> Ynm(n, m, -theta, phi) Ynm(n, m, theta, phi)
>>> Ynm(n, m, theta, -phi) exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
For specific integers \(n\) and \(m\) we can evaluate the harmonics to more useful expressions:
>>> simplify(Ynm(0, 0, theta, phi).expand(func=True)) 1/(2*sqrt(pi))
>>> simplify(Ynm(1, -1, theta, phi).expand(func=True)) sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))
>>> simplify(Ynm(1, 0, theta, phi).expand(func=True)) sqrt(3)*cos(theta)/(2*sqrt(pi))
>>> simplify(Ynm(1, 1, theta, phi).expand(func=True)) -sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))
>>> simplify(Ynm(2, -2, theta, phi).expand(func=True)) sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))
>>> simplify(Ynm(2, -1, theta, phi).expand(func=True)) sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))
>>> simplify(Ynm(2, 0, theta, phi).expand(func=True)) sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))
>>> simplify(Ynm(2, 1, theta, phi).expand(func=True)) -sqrt(30)*exp(I*phi)*sin(2*theta)/(8*sqrt(pi))
>>> simplify(Ynm(2, 2, theta, phi).expand(func=True)) sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi))
We can differentiate the functions with respect to both angles:
>>> from sympy import Ynm, Symbol, diff >>> from sympy.abc import n,m >>> theta = Symbol("theta") >>> phi = Symbol("phi")
>>> diff(Ynm(n, m, theta, phi), theta) m*cot(theta)*Ynm(n, m, theta, phi) + sqrt((-m + n)*(m + n + 1))*exp(-I*phi)*Ynm(n, m + 1, theta, phi)
>>> diff(Ynm(n, m, theta, phi), phi) I*m*Ynm(n, m, theta, phi)
Further we can compute the complex conjugation:
>>> from sympy import Ynm, Symbol, conjugate >>> from sympy.abc import n,m >>> theta = Symbol("theta") >>> phi = Symbol("phi")
>>> conjugate(Ynm(n, m, theta, phi)) (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
To get back the well known expressions in spherical coordinates, we use full expansion:
>>> from sympy import Ynm, Symbol, expand_func >>> from sympy.abc import n,m >>> theta = Symbol("theta") >>> phi = Symbol("phi")
>>> expand_func(Ynm(n, m, theta, phi)) sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*exp(I*m*phi)*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))
References
-
sympy.functions.special.spherical_harmonics.Ynm_c(n, m, theta, phi)[source]¶ Conjugate spherical harmonics defined as
\[\overline{Y_n^m(\theta, \varphi)} := (-1)^m Y_n^{-m}(\theta, \varphi).\]Examples
>>> from sympy import Ynm_c, Symbol, simplify >>> from sympy.abc import n,m >>> theta = Symbol("theta") >>> phi = Symbol("phi") >>> Ynm_c(n, m, theta, phi) (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi) >>> Ynm_c(n, m, -theta, phi) (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
For specific integers \(n\) and \(m\) we can evaluate the harmonics to more useful expressions:
>>> simplify(Ynm_c(0, 0, theta, phi).expand(func=True)) 1/(2*sqrt(pi)) >>> simplify(Ynm_c(1, -1, theta, phi).expand(func=True)) sqrt(6)*exp(I*(-phi + 2*conjugate(phi)))*sin(theta)/(4*sqrt(pi))
References
-
class
sympy.functions.special.spherical_harmonics.Znm(**kwargs)[source]¶ Real spherical harmonics defined as
\[\begin{split}Z_n^m(\theta, \varphi) := \begin{cases} \frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &\quad m > 0 \\ Y_n^m(\theta, \varphi) &\quad m = 0 \\ \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &\quad m < 0 \\ \end{cases}\end{split}\]which gives in simplified form
\[\begin{split}Z_n^m(\theta, \varphi) = \begin{cases} \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &\quad m > 0 \\ Y_n^m(\theta, \varphi) &\quad m = 0 \\ \frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &\quad m < 0 \\ \end{cases}\end{split}\]Examples
>>> from sympy import Znm, Symbol, simplify >>> from sympy.abc import n, m >>> theta = Symbol("theta") >>> phi = Symbol("phi") >>> Znm(n, m, theta, phi) Znm(n, m, theta, phi)
For specific integers n and m we can evaluate the harmonics to more useful expressions:
>>> simplify(Znm(0, 0, theta, phi).expand(func=True)) 1/(2*sqrt(pi)) >>> simplify(Znm(1, 1, theta, phi).expand(func=True)) -sqrt(3)*sin(theta)*cos(phi)/(2*sqrt(pi)) >>> simplify(Znm(2, 1, theta, phi).expand(func=True)) -sqrt(15)*sin(2*theta)*cos(phi)/(4*sqrt(pi))
References
Tensor Functions¶
-
sympy.functions.special.tensor_functions.Eijk(*args, **kwargs)[source]¶ Represent the Levi-Civita symbol.
This is a compatibility wrapper to
LeviCivita().See also
-
sympy.functions.special.tensor_functions.eval_levicivita(*args)[source]¶ Evaluate Levi-Civita symbol.
-
class
sympy.functions.special.tensor_functions.LeviCivita(**kwargs)[source]¶ Represent the Levi-Civita symbol.
Explanation
For even permutations of indices it returns 1, for odd permutations -1, and for everything else (a repeated index) it returns 0.
Thus it represents an alternating pseudotensor.
Examples
>>> from sympy import LeviCivita >>> from sympy.abc import i, j, k >>> LeviCivita(1, 2, 3) 1 >>> LeviCivita(1, 3, 2) -1 >>> LeviCivita(1, 2, 2) 0 >>> LeviCivita(i, j, k) LeviCivita(i, j, k) >>> LeviCivita(i, j, i) 0
See also
-
class
sympy.functions.special.tensor_functions.KroneckerDelta(**kwargs)[source]¶ The discrete, or Kronecker, delta function.
- Parameters
i : Number, Symbol
The first index of the delta function.
j : Number, Symbol
The second index of the delta function.
Explanation
A function that takes in two integers \(i\) and \(j\). It returns \(0\) if \(i\) and \(j\) are not equal, or it returns \(1\) if \(i\) and \(j\) are equal.
Examples
An example with integer indices:
>>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> KroneckerDelta(1, 2) 0 >>> KroneckerDelta(3, 3) 1
Symbolic indices:
>>> from sympy.abc import i, j, k >>> KroneckerDelta(i, j) KroneckerDelta(i, j) >>> KroneckerDelta(i, i) 1 >>> KroneckerDelta(i, i + 1) 0 >>> KroneckerDelta(i, i + 1 + k) KroneckerDelta(i, i + k + 1)
See also
References
-
classmethod
eval(i, j, delta_range=None)[source]¶ Evaluates the discrete delta function.
Examples
>>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy.abc import i, j, k
>>> KroneckerDelta(i, j) KroneckerDelta(i, j) >>> KroneckerDelta(i, i) 1 >>> KroneckerDelta(i, i + 1) 0 >>> KroneckerDelta(i, i + 1 + k) KroneckerDelta(i, i + k + 1)
# indirect doctest
-
property
indices_contain_equal_information¶ Returns True if indices are either both above or below fermi.
Examples
>>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, q).indices_contain_equal_information True >>> KroneckerDelta(p, q+1).indices_contain_equal_information True >>> KroneckerDelta(i, p).indices_contain_equal_information False
-
property
is_above_fermi¶ True if Delta can be non-zero above fermi.
Examples
>>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_above_fermi True >>> KroneckerDelta(p, i).is_above_fermi False >>> KroneckerDelta(p, q).is_above_fermi True
See also
-
property
is_below_fermi¶ True if Delta can be non-zero below fermi.
Examples
>>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_below_fermi False >>> KroneckerDelta(p, i).is_below_fermi True >>> KroneckerDelta(p, q).is_below_fermi True
See also
-
property
is_only_above_fermi¶ True if Delta is restricted to above fermi.
Examples
>>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_only_above_fermi True >>> KroneckerDelta(p, q).is_only_above_fermi False >>> KroneckerDelta(p, i).is_only_above_fermi False
See also
-
property
is_only_below_fermi¶ True if Delta is restricted to below fermi.
Examples
>>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, i).is_only_below_fermi True >>> KroneckerDelta(p, q).is_only_below_fermi False >>> KroneckerDelta(p, a).is_only_below_fermi False
See also
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property
killable_index¶ Returns the index which is preferred to substitute in the final expression.
Explanation
The index to substitute is the index with less information regarding fermi level. If indices contain the same information, ‘a’ is preferred before ‘b’.
Examples
>>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> j = Symbol('j', below_fermi=True) >>> p = Symbol('p') >>> KroneckerDelta(p, i).killable_index p >>> KroneckerDelta(p, a).killable_index p >>> KroneckerDelta(i, j).killable_index j
See also
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property
preferred_index¶ Returns the index which is preferred to keep in the final expression.
Explanation
The preferred index is the index with more information regarding fermi level. If indices contain the same information, ‘a’ is preferred before ‘b’.
Examples
>>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> j = Symbol('j', below_fermi=True) >>> p = Symbol('p') >>> KroneckerDelta(p, i).preferred_index i >>> KroneckerDelta(p, a).preferred_index a >>> KroneckerDelta(i, j).preferred_index i
See also
