• Complex function
• logarithm
  • branch of logarithm
  • derivative of logarithm
• exponential
  • derivative of exponential
• trigonometric function
• hyperbolic function
• inverse trigonometric function
  • derivative of inverse trigonometric function
• Contour integral of power function on circle

Complex function

logarithm

$$\begin{gathered} \log z=\ln r+i\theta \\ \text{Log }z=\ln r+i\Theta \end{gathered}$$

where $\Theta =\text{Arg }z$

• branch point $z=0$
• $e^{\log z}=z$
• $\log e^z\not\equiv z$
• multi-valued function $\log$
• $\log z_1{z}_2=\log z_1+\log z_2$
• $\text{Log }{z}_1{z}_2\not\equiv \text{Log }{z}_1+\text{Log }{z}_2$

branch of logarithm

introduce $\alpha$ s.t.

$$r>0,\alpha <\theta <\alpha +2\theta$$

• branch cut $\theta =\alpha$, not continuous across
• principal branch $\alpha =-\pi$, branch cut for principal value
• branch point $z=0$, point common to all branch cut

derivative of logarithm

$$\frac{d}{dz}\log z=\frac{1}{z}$$

• holomorphic

exponential

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

• nonzero
• periodic, $e^{z+i2\pi k}=e^z$
• entire function
• $(z_1{z}_2{)}^c\not\equiv z_1^c{z}_2^c$

derivative of exponential

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

trigonometric function

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

• entire

• $\sin (iz)=i\sinh z,\cos (iz)=\cosh z$

  $$\begin{gathered} \Rightarrow \sin z=\sin x\cosh y+i\cosh x\sin y \\ \Rightarrow |\sin z{|}^2={\sin }^{2}x+{\sinh }^{2}y \\ \Rightarrow \sin z=0\iff z=\pi k,k\in \mathbb{Z} \end{gathered}$$

hyperbolic function

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

• $\sinh (iz)=i\sin z$

  $$\begin{gathered} \Rightarrow |\sinh z{|}^2=|i\sin (y-ix)|={\sinh }^{2}x+{\sin }^{2}y \\ \Rightarrow \sinh z=0\iff z=i\pi k,k\in \mathbb{Z} \end{gathered}$$

inverse trigonometric function

$$\begin{gathered} {\sin }^{-1}z=-i\log \left(iz+\sqrt{1-z^2}\right) \\ {\cos }^{-1}z=-i\log \left(z+i\sqrt{1-z^2}\right) \end{gathered}$$

• $\log ,\sqrt{}$ are multi-valued
• holomorphic in branch

derivative of inverse trigonometric function

$$\begin{gathered} \frac{d}{dz}{\sin }^{-1}z=\frac{1}{\sqrt{1-z^2}} \\ \frac{d}{dz}{\cos }^{-1}z=-\frac{1}{\sqrt{1-z^2}} \end{gathered}$$

Contour integral of power function on circle

$${\int }_{C_R}{z}^{n}dz=\{\begin{array}{ll}0& n\ne -1\\ 2\pi i& n=-1\end{array}$$