jäägiteoreem

Jäägiteoreem, also known as the residue theorem, is a fundamental result in complex analysis. It provides a powerful method for evaluating complex contour integrals. The theorem relates the integral of a complex function around a closed curve to the sum of the residues of the function at the singularities enclosed by the curve.

A residue of a function at an isolated singularity is a specific coefficient in the Laurent series expansion of the function around that point. The Jäägiteoreem states that if f(z) is analytic inside and on a simple closed contour C, except for a finite number of isolated singularities z_1, z_2, ..., z_n inside C, then the integral of f(z) along C is equal to 2πi times the sum of the residues of f(z) at these singularities.

The theorem is incredibly useful because it often simplifies the evaluation of integrals that would otherwise be very difficult to compute directly. It has wide-ranging applications in various fields of mathematics, physics, and engineering, including solving differential equations, analyzing electrical circuits, and studying quantum field theory. The calculation of residues is a key step in applying the theorem, and various techniques exist for finding them, depending on the nature of the singularity.