SKEDSOFT

Evaluation of Definite Integrals:

One of the most classical and fascinating applications of the calculus of residues is the calculation of definite (usually improper) real integrals. It is an over-simplification to call these calculations, taken together, a “technique”: it is more like a collection of techniques. We present several instances of the method.

Basic Example of the Indefinite Integral:

To evaluate;

we “complexify” the integrand to f(z) = 1/(1 z⁴) and consider the integral,

by the calculation that we are about to do. Assume that R > 1. Define

Call these two curves, taken together

Now we set U = C,

the points P₁, P₂, P₃, P₄ are the poles of 1/[1 z⁴]. Thus f(z) = 1/(1 z⁴) is holomorphic on U \ {P₁, . . . , P₄} and the Residue Theorem applies.