Holomorphic

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A holomorphic function $f: \mathbb{C} \to \mathbb{C}$ is a differentiable complex function. That is, just as in the real case, $f$ is holomorphic at $z$ if $\lim_{h\to 0} \frac{f(z+h)-f(z)}{h}$ exists. This is much stronger than in the real case since we must allow $h$ to approach zero from any direction in the complex plane.

Cauchy-Riemann Equations

Let us break $f$ into its real and imaginary components by writing $f(z)=u(x,y)+iv(x,y)$ , where $u$ and $v$ are real functions. Then it turns out that $f$ is holomorphic at $z$ iff $u$ and $v$ have continuous partial derivatives and the following equations hold:

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

These equations are known as the Cauchy-Riemann Equations.

Analytic Functions

A related notion to that of homolorphicity is that of analyticity. A function $f:\mathbb{C}\to\mathbb{C}$ is said to be analytic at $z$ if $f$ has a convergent power series expansion on some neighborhood of $z$ . Amazingly, it turns out that a function is holomorphic at $z$ if and only if it is analytic at $z$ .

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Holomorphic

From AoPSWiki

Cauchy-Riemann Equations

Analytic Functions