Tagged: gamma-function

Differentiating the Mellin transform of the sine function with respect to its exponent evaluates a logarithmic Fresnel-type integral in terms of Euler's constant.

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.

Reflecting the interval of integration and applying Euler's reflection formula evaluates a definite integral involving a fifth root.

A homographic change of variables reduces a beta-type integral with a shifted linear denominator to Euler's beta function.

A change of variables and an inversion symmetry reduce integral 3.255 from Gradshteyn and Ryzhik to a beta integral.

A contour argument gives integral representations involving the reciprocal beta function and their logarithmic consequences.

Use Ramanujan's master theorem to connect power-series coefficients with Mellin transforms of Bessel functions.

Derive Kummer's Fourier series for the logarithm of the gamma function, then use it to evaluate Vardi's integral.