Complex Analysis Handbook/Functions/Derivative

$\frac{dw}{dz}$	$=\underset{\mathrm{\Delta }z\to 0}{\lim }\frac{\mathrm{\Delta }w}{\mathrm{\Delta }z}$
	$=\underset{\mathrm{\Delta }x\to 0,\mathrm{\Delta }y\to 0}{\lim }\frac{u(x+\mathrm{\Delta }x,y+\mathrm{\Delta }y)+iv(x+\mathrm{\Delta }x,y+\mathrm{\Delta }y)-(u(x,y)+iv(x,y))}{\mathrm{\Delta }x+i\mathrm{\Delta }y}$

If $\mathrm{\Delta }z$ is real (it means $\mathrm{\Delta }y=0$), performing some calculations, we obtain:

$\frac{dw}{dz}=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$

In the same manner, if $\mathrm{\Delta }z$ is imaginary ($\mathrm{\Delta }x=0$):

$\frac{dw}{dz}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}$

We conclude:

$\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}$

The last equation is called Cauchy Riemann' test.

Analytic Function

If $w=f(z)$ posseses a derivative at $z=z_0$ and at every point in a neighbourhood of $z_0$, $f(z)$ is analytic at $z_0$ and $z_0$ is a regular point for $w$.

Properties of Analytic Functions

• The real and imaginary parts of an analytic function satisfy Laplace's Equation:

$\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}=0$

Proof:

Let $w=u(x,y)+iv(x,y)$ be analytic at some point $z_0$. We have: $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$ (Cauchy Riemann's formula)

$\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}v}{\partial y\partial x}$ and $\frac{{\partial }^{2}u}{\partial y^2}=-\frac{{\partial }^{2}v}{\partial x\partial y}$

so $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$