Analytic Function

The theory of complex functions mainly deal with analytical functions.

$f^{'} (z) = \frac{d}{d z} f (z) = lim_{Δ z \to 0} \frac{f (z + Δ z) - f (z)}{(z + Δ z) - z} = lim_{Δ z \to 0} \frac{f (z + Δ z) - f (z)}{Δ z}$

Let $Δ z = Δ x$ . $f^{'} (z) = lim_{Δ x \to 0} \frac{f (z + Δ x) - f (z)}{Δ x} = lim_{Δ x \to 0} \frac{u (x + Δ x, y) + i v (x + Δ x, y) - u (x, y) - i v (x, y)}{Δ x} = lim_{Δ x \to 0} \frac{[u (x + Δ x, y) - u (x, y)] + i [v (x + Δ x, y) - v (x, y)]}{Δ x} = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}$

Let $Δ z = i Δ y$ . $f^{'} (z) = lim_{Δ y \to 0} \frac{f (z + i Δ y) - f (z)}{i Δ y} = lim_{Δ y \to 0} \frac{u (x, y + Δ y) + i v (x, y + Δ y) - u (x, y) - i v (x, y)}{i Δ y} = lim_{Δ y \to 0} \frac{[u (x, y + Δ y) - u (x, y)] + i [v (x, y + Δ y) - v (x, y)]}{i Δ y} = - i \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}$

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

Definition

A function $f$ is analytic if $f$ has a derivative at $z$ and in the neighborhood of $z$ . (I.e. in a region $R \subset C$ )

Also called entire or holomorphic.

Entire means that $f$ is holomorphic in $R = C$ .

Meromorphic means that $f$ is holomorphic in $R = C$ except for a set of isolated poles.

Cauchy-Riemann Conditions

Thus, $\frac{\partial u}{\partial x} = i \frac{\partial v}{\partial y}$ and $- \frac{\partial v}{\partial x} = \frac{\partial u}{\partial y}$ This is called the Cauchy-Riemann Conditions.

Aside

If you can write a function $f$ solely in terms of $z$ (not $z^{*}$ ) (e.g. $f (z) = 1 / z, \sin z, z^{2}$ ) then it is differentiable everywhere it is finite. An example of one that is not, $| z |^{2} = z z^{2}$ is not entire.

If it is differentiable then all of the derivative rules from before hold. I.e. $\frac{d}{d z} z^{n} = n z^{n - 1}$ etc.