Contour Integration

Integrating a complex power around a keyhole cut along the negative real axis evaluates a rational integral involving the sine of a logarithm.

A trigonometric substitution and integration around the right half of the unit disk reduce an inverse-sine integral to a beta function.

Rotating the positive real axis to the positive imaginary axis via an indented quarter-circle contour evaluates the Mellin transforms of sine and cosine.

Residue calculus on a rectangular contour shows that the hyperbolic secant is its own Fourier transform up to scaling.

Integrating a meromorphic function around a horizontal strip of height two pi evaluates a real integral mixing exponential and rational factors.

A contour-integral approach to an infinite trigonometric series, with an application of equidistribution.