Complex Variables
De Moivre Powers
z = x + iy = eiθ = cos θ + i sin θ (cos θ + i sin θ)n = cos nθ + i sin nθ
1 (cos θ + i sin θ)−n = cos nθ − i sin θ
z̄ = = x − iy = e−iθ = cos θ − i sin θ
z
θ θ + 2πk θ + 2πk 2 cos nθ = z n + z −n
ei n = cos + i sin 2i sin nθ = z n − z −n
n n
Trigonometric Hyperbolic
eiθ − e−iθ
1 1 eθ − e−θ
=z = sin θ = = z− sinh θ =
2i 2i z 2
e + e−iθ
iθ eθ + e−θ
1 1
<z = cos θ = = z+ cosh θ =
2 2 z 2
sin θ sinh θ
tan θ = tanh θ =
cos θ cosh θ
Complex Equivalences
cos iz = cosh z
sin iz = i sinh z cosh iz = cos z
√
sinh iz = i sin z cosh x = k → x = ln(k ± k 2 − 1), where(k > 1)
Hyperbolic Identities
cosh(a + b) = cosh a cosh b + sinh a sinh b
cosh z = cosh x cos y + i sinh x sin y
Complex Roots Roots of Unity
1 1 1 1 θ
in
ω = z n= r (cos θ + i sin θ) = r e
n n n
1 = cos 2πk + i sin 2πk = ei2πk
θ + 2πk θ + 2πk
1
2 n−1 2πi
= r n cos + i sin 1, α, α , . . . , α with α = exp
n n n
Author: Martin Blais, 2009. This work is licensed under the Creative Commons “Attribution - Non-Commercial - Share-Alike” license.