Complex Analysis: Theory and Applications · Problem 18

Complex Integration — Howell & Matthews

Mathematics — Complex Analysis · complex-definite-integrals-partial-fractions

Problem

Using partial fraction decomposition, show that if

z

lies in the right half-plane and

C

is the line segment joining

0

z

, then

\int_C \frac{1}{\xi^2+1}\,d\xi = \operatorname{Arctan}(z) = \frac{i}{2}\operatorname{Log}(z+i) - \frac{i}{2}\operatorname{Log}(z-i) + \frac{\pi}{2}.