Complex function

logarithm

$lo g z = ln r + i θ Log z = ln r + i Θ$

where $Θ = Arg z$

branch point $z = 0$
$e^{l o g z} = z$
$lo g e^{z} \neq \equiv z$
multi-valued function $lo g$
$lo g z_{1} z_{2} = lo g z_{1} + lo g z_{2}$
$Log z_{1} z_{2} \neq \equiv Log z_{1} + Log z_{2}$

branch of logarithm

introduce $α$ s.t.

$r > 0, α < θ < α + 2 θ$

branch cut $θ = α$ , not continuous across
principal branch $α = - π$ , branch cut for principal value
branch point $z = 0$ , point common to all branch cut

derivative of logarithm

$\frac{d}{d z} lo g z = \frac{1}{z}$

holomorphic

exponential

$e^{z} = e^{x} e^{i y} = e^{x} cos y + i e^{x} sin y$

nonzero
periodic, $e^{z + i 2 π k} = e^{z}$
entire function
$(z_{1} z_{2})^{c} \neq \equiv z_{1}^{c} z_{2}^{c}$

derivative of exponential

$\frac{d}{d z} c^{z} = c^{z} Log c$

trigonometric function

$sin z = \frac{e ^{i z} - e ^{- i z}}{2 i}, cos z = \frac{e ^{i z} + e ^{- i z}}{2}$

entire
$sin (i z) = i sinh z, cos (i z) = cosh z$

$\Rightarrow sin z = sin x cosh y + i cosh x sin y \Rightarrow ∣ sin z ∣^{2} = sin^{2} x + sinh^{2} y \Rightarrow sin z = 0 \Leftrightarrow z = π k, k \in Z$

hyperbolic function

$sinh z = \frac{e ^{z} - e ^{- z}}{2}, cosh z = \frac{e ^{z} + e ^{- z}}{2}$

$sinh (i z) = i sin z$

$\Rightarrow ∣ sinh z ∣^{2} = ∣ i sin (y - i x) ∣ = sinh^{2} x + sin^{2} y \Rightarrow sinh z = 0 \Leftrightarrow z = iπ k, k \in Z$

inverse trigonometric function

$sin^{- 1} z = - i lo g (i z + 1 - z^{2}) cos^{- 1} z = - i lo g (z + i 1 - z^{2})$

$lo g,$ are multi-valued
holomorphic in branch

derivative of inverse trigonometric function

$\frac{d}{d z} sin^{- 1} z = \frac{1}{1 - z ^{2}} \frac{d}{d z} cos^{- 1} z = - \frac{1}{1 - z ^{2}}$

Contour integral of power function on circle

$\int_{C_{R}} z^{n} d z = {0 2 π i n \neq = - 1 n = - 1$

Last edited: 2024-09-13