Complex Analysis: Theory and Applications · Problem 8.6.10

Conformal Mapping — Howell & Matthews

Mathematics — Complex Analysis · contour-integrals-residues

Problem

Carry out the following computations: (a) For

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

, show that

\mathrm{Res}[f, -1] = -\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i

. (b) Use part (a) and

\alpha = \dfrac{1}{3}

to verify the computation

\dfrac{2\pi i}{1 - e^{i\alpha 2\pi}} \mathrm{Res}[f, -1] = \dfrac{2\sqrt{3}}{3}\pi

. (c) Can you conclude that

\displaystyle\text{P.V.} \int_0^\infty \frac{x^{\frac{1}{3}}}{x^3(x+1)} \, dx = \frac{2\sqrt{3}}{3}\pi

? Justify your answer.