Eikonal Blog

2010.01.04

Mellin-Barnes integrals

Filed under: mathematics — Tags: , — sandokan65 @ 20:02

\int_{-i\infty}^{+i\infty} \Gamma(\alpha+s) \Gamma(\beta+s) \Gamma(\gamma-s) \Gamma(\delta-s) ds = 2\pi i \frac{\Gamma(\alpha+\gamma)\Gamma(\beta+\gamma)\Gamma(\alpha+\delta)\Gamma(\beta+\delta)}{\Gamma(\alpha \beta+\gamma+\delta)}

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Integrals with Gamma functions

Filed under: mathematics — Tags: , , — sandokan65 @ 20:00
  • \int_0^\infty \frac{\cosh(2yt)}{(\cosh(t))^{2x}} dt = 2^{2x-2} \frac{\Gamma(x+y)\Gamma(x-y)}{\Gamma(2x)}, for \Re x > |\Re y|.
  • \int_{-i\infty}^{+i\infty} \frac{e^{its}ds}{\Gamma(\mu+s)\Gamma(\nu-s)} = \theta(\pi - |t|) \frac{2(\cos(\frac{t}2))^{\mu+\nu-2} e^{\frac{1}{2} i t (\nu-\mu)}}{\Gamma(\mu+\nu-1)}.

Sources:

  • “The Functions of Mathematical Physics” by Harry Hochstadt

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