# MATH3401

Lecture 16

### Derivative of $$\log z$$

$$\ds \frac{d}{dz}\log z$$:   Consider $$|z|>0$$ and recall

\begin{align} \log z & = \ln |z| + i \,\arg(z) \\ & = \ln r + i \theta. \end{align}

$$\implies u = \ln r, v = \theta$$

$$\implies u_r = \ds \frac{1}{r}, u_{\theta} = 0 , v_r =0, v_{\theta} = 1.$$

### Derivative of $$\log z$$

C/R in polar coordinates: \begin{align} ru_r & = v_{\theta} \\ u_{\theta} & = -ru_r \end{align}

Satisfies sufficient conditions for complex differentiability on any subset of $$\C_*$$, such that $$\alpha \lt \theta \lt \alpha + 2 \pi$$, with $$\alpha$$ fixed in $$\R$$.

### Derivative of $$\log z$$

So if we specify a branch, we have that $$\log$$ is differentiable and expression (6) in Lecture 15 $$\implies$$

$$\ds \frac{d}{dz}\log z$$ $$= e^{-i\theta} \left( u_r + i v_r \right)$$ $$= e^{-i\theta}\ds \frac{1}{r}$$ $$= \ds \frac{1}{re^{i\theta}}$$ $$= \ds \frac{1}{z}$$

### Derivative of $$\log z$$

E. g. $$\ds \frac{d}{dz}\Log\, z$$ $$\ds = \frac{1}{z}$$

for $$-\pi \lt \Arg \, z \lt \pi$$, $$|z|> 0$$.

### Derivative of $$z^c$$

For $$f(z)= z^c$$, with $$c\in \C_*$$ fixed, defined on $$\C_*$$:

$f(z) = \exp\left( c \log z \right)$

$$f'(z) = \exp\left( c \log z \right) \cdot \ds \frac{c}{z}$$     $$(*)$$

$$\qquad = z^c \cdot \ds \frac{c}{z}$$ $$= c z^{c-1}$$      $$(**)$$

### Derivative of $$z^c$$

$$(**)$$ is valid on any domain of the form $\left\{ z: |z|>0, \alpha \lt \arg\, z \lt \alpha + 2\pi\right\}$

(due to needing to pick a branch of $$\log z$$ in $$(*)$$).

📝Remark: Try for $$g(z) = c^z$$

### Notation from Real Analysis

$$\Omega \subseteq \R^n$$

$$*$$ $$\mathcal C\left( \Omega \right)$$ $$= \mathcal C^0\left( \Omega \right)$$

$$\quad\quad\;\, = \left\{ \text{continuous functions}: \Omega \ra \R \right\}$$.

$$*$$ $$\mathcal C^k\left( \Omega \right)$$ $$= \left\{ \text{functions } f:\Omega \ra \R:\right.$$ $$\qquad \quad\left. f \text{ and all its derivatives/partial derivatives } \right.$$ $$\qquad \quad\left. \text{of order } 0,1,\ldots, k \text{ exist and are continuous} \right\}$$.

Note: $$0$$th order derivative of $$f$$ is $$f$$.

### Notation from Real Analysis

$$*$$ $$\mathcal C^{\infty}\left( \Omega \right)$$ $$= \left\{ \text{functions } f:\Omega \ra \R:\right.$$ $$\qquad \quad\left. f \text{ and all its derivatives/partial derivatives } \right.$$ $$\qquad \quad\left. \text{of all orders exist and are continuous} \right\}$$.

a.k.a. Smooth functions

### Notation from Real Analysis

$$*$$ $$f\in \mathcal C^{\omega} \left( \Omega \right)$$ says at every $$x_0\in \Omega$$:

1. $$f$$ has a power series expansion about $$x_0$$. Namely, its Taylor series.
2. $$f$$ is given by its power series expansion, i.e., power series converges to $$f$$ on some neighbourhood of $$x_0$$. a.k.a real analytic functions.

Note: (i) $$\Rightarrow f\in C^{\infty}(\Omega)$$ ,    (i) $$\nRightarrow$$ (ii) in $$\R^{n}$$.

### Notation from Real Analysis

Consider $f(x) = \left\{ \begin{array}{lr} e^{-1/x^2} & x > 0\\ 0 & x \le 0 \end{array} \right.$

📝Check:  $$f^{(n)}(x)$$ exists for every $$x\neq 0$$ , and $$f^{(n)}(0) = 0$$ $$\,\forall n\in \N$$, and $$f^{(n)}$$ is continuous on $$\R$$. Then Taylor series for $$f$$ at $$0$$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n \equiv 0.$

$$f\in \mathcal C^{\infty}(\R), \quad f\notin \mathcal C^{\omega}(\R)$$.

### Notation from Real Analysis

$$\quad$$

$$\mathcal C^{\omega}$$ $$\subsetneq \mathcal C^{\infty}$$ $$\subsetneq \ldots \subsetneq \mathcal C^{1337} \subsetneq \mathcal C^{1336}$$

$$\subsetneq \ldots \subsetneq \mathcal C^{1} \subsetneq \mathcal C^{0}$$.

$$\quad$$