# MATH3401

Lecture 15

### Cauchy-Riemann (C/R)

in polar coordinates

$$z = x+iy=r e^{i\theta }$$,   $$x = r\cos \theta$$, $$y = r \sin \theta$$.

Chain rule implies

$$u_r = u_x \cos \theta + u_y \sin \theta \quad \quad (1)$$

$$u_{\theta} = -u_x r \sin \theta + u_y r \cos \theta \quad (2)$$

$$v_r = v_x \cos \theta + v_y \sin \theta \quad \quad (3)$$

$$v_{\theta} = -v_x r \sin \theta + v_y r \cos \theta \quad (4)$$

### Cauchy-Riemann

in polar coordinates

Combine with C/R $$\implies$$ C/R in polar coordinates:

\begin{align} ru_r & = v_{\theta} \\ u_{\theta} & = -ru_r \end{align}

### Cauchy-Riemann

in polar coordinates

Useful: (recall (⚛️), Lecture 14)

If $$f'$$ exists then \begin{align} f' & = u_x + i v_x \qquad (5), \end{align} and similarly 📝 \begin{align} f'(z) & = e^{-i\theta}\left( u_r + i v_r\right) \qquad (6) \end{align}

### Another approach to C/R

Formally: change of variables $$\left(x,y\right) \mapsto \left(z,\overline{z}\right)$$

$z= x+iy, \quad \overline{z}= x-iy$

### Another approach to C/R

$$\ds \frac{\partial f}{\partial x}$$ $$= \ds \frac{\partial f}{\partial z} \frac{\partial z}{\partial x} + \frac{\partial f}{\partial \overline{z}} \frac{\partial \overline{z}}{\partial x}$$

$$\quad\; = \ds \frac{\partial f}{\partial z} + \frac{\partial f}{\partial \overline{z}} \quad (7)$$

$$\ds \frac{\partial f}{\partial y}$$ $$= \ds \frac{\partial f}{\partial z} \frac{\partial z}{\partial y} + \frac{\partial f}{\partial \overline{z}} \frac{\partial \overline{z}}{\partial y}$$

$$\quad\; = \ds i \frac{\partial f}{\partial z} - i \frac{\partial f}{\partial \overline{z}} \quad (8)$$

### Another approach to C/R

(7)$$-i$$(8) $$\implies$$ $$\ds \frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y}$$ $$= 2 \ds \frac{\partial f}{\partial z}$$ $$\implies \ds \frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right)$$

Finally (7)$$+i$$(8) $$\implies$$ $$\ds \frac{\partial}{\partial \overline{z}} = \frac{1}{2}\left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right)$$

$$\ds \frac{\partial}{\partial z}, \frac{\partial}{\partial \overline{z}}$$ are called the Wirtinger operators.

### Wirtinger operators

Example

$$f(z) = z^n = (x+iy)^n$$ with $$n \in \Z$$.

$$\ds \frac{\partial f}{\partial z}$$ $$= \ds \frac12 \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) (x+iy)^n$$ $$= \ds \frac12 n(x+iy)^{n-1}\left(1-i^2\right)$$

$$\quad \;= \ds n(x+iy)^{n-1}$$ $$= nz^{n-1}$$ $$= f'(z)$$.

$$\ds \frac{\partial f}{\partial \overline{z}} =0$$.

### Wirtinger operators

Relationship with C/R

For $$f = u+iv$$ complex differentiable:

$$\ds \frac{1}{2} \frac{\partial f}{\partial x}$$ $$= \ds \frac12 \left( u_x + iv_x \right)$$ $$= \ds \frac12 \left( v_y- iu_y \right)$$

$$\qquad \;= \ds - \frac{i}{2} \left( u_y+ iv_y \right)$$ $$= \ds - \frac{i}{2} \frac{\partial f}{\partial y}$$.

So C/R $$\iff \ds \frac{\partial f}{\partial \overline{z}} =0$$.

### Wirtinger operators

Relationship with C/R

Note also from (5):

$$\ds \frac{d f}{d z}$$ $$= u_x + iv_x$$ $$= \ds \frac{\partial f}{\partial x}$$ $$= \ds - i \frac{\partial f}{\partial y}$$ $$= \ds \frac{i}{2} \left( \frac{\partial f}{\partial x} - i\frac{\partial f}{\partial y} \right)$$ $$= \ds \frac{\partial f}{\partial z}$$.

### Wirtinger operators

Application 1

Find $$f'(z)$$ for $$f(z) = \exp z$$.

Check sufficient conditions for $$f'$$ to exist:

a) $$f= u+iv$$ $$= e^{x+iy}$$ $$= e^x \left(\cos y +i \sin y\right)$$ defined on $$\C$$.

$$u = e^x\cos y, v = e^{x}\sin y$$.   $$u_x, u_y, v_x, v_y$$ defined and continuous on $$\C$$.

b) Check C/R: $$u_x=v_y$$ & $$u_y = -v_x$$   or $$\ds \frac{\partial f}{\partial \overline{z}}=0.$$

### Wirtinger operators

Application 2

When is $$g(z) = |z|^2$$ differentiable?

Note: $$g(z)=z\overline{z} = x^2+y^2$$.

C/R: $$\ds \frac{\partial g}{\partial \overline{z}}= 0$$ $$\implies z = 0$$

So $$g$$ cannot be differentiable for $$z\neq 0$$ because C/R is necessary for differentiability.

### Wirtinger operators

Application 2

At $$z=0$$ check sufficient conditions:

$$u,v$$ are defined on $$\C$$. $$u_x,u_y,v_x,v_y$$ are defined and continuous on $$\C$$ (a neighbourhood of $$0$$).

C/R hold. So $$g'(0)=0$$.

📝Exercise: Go through Application 1 for $$z\mapsto 1/z$$ on $$\C_*$$

### Analytic and singular function

Definition: $$f:\Omega \ra \C$$ is analytic at $$z_0$$ if $$f$$ is differentiable on a neighbourhood of $$z_0$$.

A function is singular at $$z_0$$ if it is NOT analytic at $$z_0$$, but is analytic at some point in every neighbouhood of $$z_0$$.

### Analytic and singular function

E. g.   $$f(z)=\ds \frac{1}{z}$$ is analytic on $$\C_*$$ and singular at $$0$$ on $$B_{\varepsilon}(0)$$,   $$f$$ is analytic on $$B_{\varepsilon'}(z_0)$$, $$z_0\in B_{\varepsilon}(0)$$, $$\varepsilon'\lt |z_0|$$.

### Entire function

A function is entire if it is analytic on all of $$\C$$.

E. g. Polynomials, $$\sin,\cos, \exp, \cosh, \sinh$$.

Note: If a function is differentiable at precisely one point,
it is not analytic there, or anywhere. E. g. $$|z|^2$$.