DOKK Library

Complex Variables

Authors Martin Blais,

License CC-BY-NC-SA-4.0

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                                       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
                           iθ    −iθ                                                                           eθ + e−θ
                                               
                          e +e         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.