Isolated Singularities and Residue Theorem

First of all, I want to apologize for the names I'm going to use on this wiki, because many of them probably have different names when written in books. We'll suppose that \(\Omega \subseteq \mathbb{C}\) is an open set throughout, unless otherwise stated, and \(f \in \mathcal{H}ig(\Omegaig)\) means \(f\) is a holomorphic function in \(\Omega\). We are going to deal with integrals, series, Bernouilli numbers, Riemann zeta function, and many interesting problems, as well as many theories. It's very important to tell everything is very joined and connected inside of complex analysis, so we'll use much knowledge of complex analysis and all the branches of mathematics.

Relevant wiki: Contour Integration

If \(\gamma : [0,1] o \mathbb{C}\) is a closed piecewise smooth path and \(a otin \gammaig([0,1]ig)\), the winding number of \(\gamma\) with respect to \(a\) \((\)number of turns around \(a\) by \(\gamma\) anticlockwise) is

\[\displaystyle ext{ Ind(}\gamma , a) = rac{1}{2i \pi} \cdot \int_{0}^{1} rac{\gamma '(s)}{\gamma(s) - a} \, ds = rac{1}{2i \pi} \oint_{\gamma} rac{1}{z - a} \, dz.\]

If \(\gamma (x) = e^{2i \pi x}\) for \(x \in [0, 1],\) the winding number around \(0\) is

\[\displaystyle ext{ Ind(}\gamma , 0) = rac{1}{2i \pi} \cdot \int_{0}^{1} rac{2i \pi \cdot e^{2i \pi x}}{e^{2i \pi x}} \, dx = 1.\]

If \(\gamma_1 (x) = e^{i \pi x}\) for \(x \in [0, 1],\)

\[\displaystyle rac{1}{2i \pi} \cdot \int_{0}^{1} rac{i \pi \cdot e^{i \pi x}}{e^{i \pi x}} \, dx = rac{1}{2}.\]

Nevertheless, we can't say

\[\displaystyle ext{ Ind(}\gamma_1 , 0) = rac{1}{2}\]

because \(\gamma_1\) is not closed.

The winding number of a path \(\gamma\) with respect to \(a\) is constant in the connected component containing (or not) \(a\). This is, the winding number is 0 if \(\gamma\) doesn't turn around \(a,\) and it's constant if \(\gamma\) turns around \(a\). (Think about the winding number as the number of turns around...)

• Proposition: (generalization of the previous definition)
  If \(C_r (t) = a + re^{it}\) for \(t \in [0, 2\pi],\) then \( ext{ Ind(C}_r, z) = 1\) if \(|z - a| < r\) and \( ext{ Ind(C}_r, z) = 0\) if \(|z - a| > r.\)