Introduction to Complex Analysis

holomorphicity

A function $f:\mathbb C\rightarrow\mathbb C$ is holomorphic at the point $z\in\mathbb C$ if the limit

$$\lim_{h\to 0}\frac{f(z+h)-f(z)}{h}(h\in\mathbb C)$$

exists.

characteristic properties of holomorphic functions

• Contour integration: If f is holomorphic in $\Omega$,then for appropriate closed paths in $\Omega$

  $$\int_\gamma f(z)dz=0$$

• Regularity: If f is holomorphic, then f is indefinitely differentiable.

• Analytic continuation: If f and g are holomorphic functions in $\Omega$ which are equal in an arbitrarily small disc in $\Omega$,then f=g everywhere in $\Omega$.

• The zeta function

  $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$$

• The theta function

  $$\Theta(z|\tau)=\sum_{n=-\infty}^{\infty}e^{\pi in^2}e^{2\pi inz}$$

• Basic properties

The complex plane

Commutativity

Associativity

Distributivity

The absolute value

The triangle inequality

The complex conjugate

In polar form