03 - Exponentials, Trigonometric Functions, Logarithms, Powers

Exponential Function

$$e^z=\exp (z)=\sum _{n=0}^{\infty }\frac{z^n}{n!}$$

with

$${\exp }^{\prime }=\exp \text{ and }{e}^{iz}=\exp (iz)=\cos (z)+i\sin (z)$$

Holomorphic on all of $\mathbb{C}$ and derivatives given by term-by-term differentiation of series

Cosine Function

$$\cos (z)=\sum _{n=0}^{\infty }(-1{)}^n\frac{z^{2n}}{(2n)!}$$

with

$${\cos }^{\prime }=-\sin$$

Holomorphic on all of $\mathbb{C}$ and derivatives given by term-by-term differentiation of series

Sine Function

$$\sin (z)=\sum _{n=0}^{\infty }(-1{)}^n\frac{z^{2n+1}}{(2n+1)!}$$

with

$${\sin }^{\prime }=\cos$$

Holomorphic on all of $\mathbb{C}$ and derivatives given by term-by-term differentiation of series

Properties of Exponential and Trigonometric Functions

$$\sin (-z)=\sin (z)\cos (-z)=\cos (z)$$

with

$$\cos (z)=\frac{e^{iz}+e^{-iz}}{2}\sin (z)=\frac{e^{iz}-e^{-iz}}{2}$$

Property of Exponential Multiplication

For $z,w\in \mathbb{C}$

$$\exp (z+w)=\exp (z)\exp (w)$$

Proof

For fixed $a\in \mathbb{C}$ then
Consider

$$f(z)=\exp (z)\exp (a-z)$$

By differentiation and using product rule then

$$f^{\prime }(z)=\exp (z)\exp (a-z)-\exp (z)\exp (a-z)=0$$

So by Holomorphic functions with zero derivative are constant then

$$f(z)=f(a)=\text{constant}$$

Hence

$$\exp (z)\exp (z-a)=\exp (a)$$

Substituting $a=z+w$ gets the final result

Property of Complex Exponentials

For $x,y\in \mathbb{R}$ then

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

Log Function

Let

$$D=\mathbb{C}\setminus (-\infty ,0]$$

Define function $\mathrm{Log}:D\to \mathbb{C}$
If $z=|z|e^{i\theta }$ for $\theta \in (-\pi ,\pi ]$ then define

$$\mathrm{Log}(z):=\log |z|+i\theta$$

So $\mathrm{Log}$ is holomorphic on $D$ and

$${\mathrm{Log}}^{\prime }(z)=\frac{1}{z}$$

Proof - Derivative

As $|z|$ and $\log$ is continuous then the real part of $\mathrm{Log}$ is continuous

Need to show that $\theta =\theta (z)$ is a continuous function of $z$
By the law of cosines then

$$\cos (\theta (z+h)-\theta (z))=\frac{|z{|}^2+|z+h{|}^2-|h{|}^2}{2|z||z+h|}$$

For all $|z|>0$ we have that

$$\cos (\theta (z+h)-\theta (z))\to 1\text{ as }h\to 0$$

Since $z\in D$ and $\theta \in (-\pi ,\pi )$ then this implies that

$$\theta (z+h)\to \theta (z)$$

Hence $\mathrm{Log}$ is continuous

Exclude negative real line from $D$ as argument would be $\pi$ so $\theta$ would not be continuous

By Simplified inverse function theorem then $\mathrm{Log}$ is differentiable and it’s inverse if

$$\frac{1}{{\exp }^{\prime }(\mathrm{Log}(z))}=\frac{1}{\exp (Log(z))}=\frac{1}{z}$$

Since this works for any $z\in D$ then $\mathrm{Log}$ is holomorphic in $D$ and it’s derivative is

$$\frac{1}{z}$$

Properties of Log and Powers

For $\alpha ,z,w\in \mathbb{C}$ with $z,w\ne 0$ then

$$\begin{array}{rl}[\log z]+[\log w]& =[\log zw]\\ [z^{\alpha }][w^{\alpha }]& =[(zw{)}^{\alpha }]\end{array}$$

Note that in general $[z^{\alpha }][z^{\beta }]\ne [z^{\alpha +\beta }]$

Proof

Let $\theta$ and $\varphi$ be some values of $\arg (z)$ and $\arg (w)$ then

$$\begin{array}{rl}[\log z]& =\{\log |z|+i\theta +2\pi in:n\in \mathbb{Z}\}\\ [\log w]& =\{\log |w|+i\varphi +2\pi ik:k\in \mathbb{Z}\}\end{array}$$

By adding these sets term by term then

$$\begin{array}{rl}[\log z]+[\log w]& =\{\log |z|+\log |w|+i(\theta +\varphi )+2\pi i(k+n):k,n\in \mathbb{Z}\}\\ & =\{\log |zw|+i(\theta +\varphi )+2\pi im:m\in \mathbb{Z}\}\\ & =[\log (zw)]\end{array}$$

Similarly

$$\begin{array}{rl}[z^{\alpha }]& =\{|z{|}^{\alpha }\exp (ia\theta +i\alpha 2\pi in)\}\\ [w^{\alpha }]& =\{|w{|}^{\alpha }\exp (ia\varphi +i\alpha 2\pi ik)\}\end{array}$$

Multiplying these sets term by term then

$$\begin{array}{rl}[z^{\alpha }][w^{\alpha }]& =\{|zw{|}^{\alpha }\exp (i\alpha (\theta +\varphi )+i\alpha 2\pi i(n+k)):k,n\in \mathbb{Z}\}\\ & =\{|zw{|}^{\alpha }\exp (i\alpha (\theta +\varphi )+i\alpha 2\pi im):m\in \mathbb{Z}\}\\ & =[(zw{)}^{\alpha }]\end{array}$$