02 - Holomorphic Criterion

Holomorphic Criterion

Suppose that $U\subseteq \mathbb{C}$ is open with function $f:U\to \mathbb{C}$
Let the components of $f$ be $(u,v)$ where $u,v:{\mathbb{R}}^2\to \mathbb{R}$

Suppose that all four partial derivatives ${\partial }_{x}u,{\partial }_{y}u,{\partial }_{x}v,{\partial }_{y}v$ exist and continuous in $U$
and that they satisfy the Cauchy-Riemann Equations then

$$f\text{ is holomorphic on }U\text{ with derivative }{\partial }_{x}u+i{\partial }_{x}v$$

Proof Differentiability Criterion then As the partial derivatives are continuous in $U$ then $f$ is real differentiable

By

Hence for $z_0\in U$

$$f(z)=f(z_0)+L(z-z_0)+R(z)$$

where $L=d{f}_{z_0}$ is a real linear transformation with matrix (in standard basis)

$$L=d{f}_{z_0}=\left(\begin{array}{cc}{\partial }_{x}u(x_0,y_0)& {\partial }_{y}u(x_0,y_0)\\ {\partial }_{x}v(x_0,y_0)& {\partial }_{y}v(x_0,y_0)\end{array}\right)$$

and $\frac{|R|}{|z-z_0|}\to 0$

Using the Cauchy-Riemann Equations then it can be rewritten as

$$L=d{f}_{z_0}=\left(\begin{array}{cc}{\partial }_{x}u(x_0,y_0)& -{\partial }_{x}v(x_0,y_0)\\ {\partial }_{x}v(x_0,y_0)& {\partial }_{x}u(x_0,y_0)\end{array}\right)$$

From the Cauchy-Riemann Equations - Alternative Proof then $L$ corresponds to
Complex Multiplication by ${\partial }_{x}u(x_0,y_0)+i{\partial }_{x}v(x_0,y_0)$

Hence

$$f(z)=f(z_0)+({\partial }_{x}u(x_0,y_0)+i{\partial }_{x}v(x_0,y_0))(z-z_0)+R(z)$$

Exactly the same formula as Expansion of a function around a point