Complex Analysis: Theory and Applications · Problem 8.6.16

Conformal Mapping — Howell & Matthews

Mathematics — Complex Analysis · contour-integrals-residues

Problem

Use residues to compute the following integrals:

\text{P.V.} \int_0^\infty \frac{x^{\frac{1}{3}} \ln x}{x^2+1} \, dx \quad \text{and} \quad \text{P.V.} \int_0^\infty \frac{x^{\frac{1}{3}}}{x^2+1} \, dx.

Hint: Use the complex integrand

f(z) = \dfrac{z^{\frac{1}{3}} \operatorname{Log} z}{z^2+1}