Complex Analysis explores the properties and applications of complex numbers and functions, emphasizing their geometric interpretations and analytical techniques.
Singularities are points where a function fails to be analytic. They are like 'holes' or 'spikes' in the complex plane.
The residue theorem allows us to evaluate complex integrals by looking at the residues (special coefficients) at singularities inside the contour: \[ \oint_{\gamma} f(z) , dz = 2\pi i \sum \text{Residues inside } \gamma \]
This theorem makes difficult real integrals easy and helps solve problems in physics and engineering.
Calculating \( \oint_{|z|=1} \frac{1}{z} dz \) gives \( 2\pi i \) using the residue at \( z=0 \).
The integral of \( \frac{e^z}{z^2} \) around a small loop around the origin can be found using the residue at \( z=0 \).
Singularities are special points in the complex plane, and the residue theorem uses them to compute complex integrals efficiently.