Eikonal Blog

2010.01.04

Logarithmic Gamma function

Filed under: mathematics — Tags: , — sandokan65 @ 20:17

Definition:

    \psi(z) :\equiv d_z \ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}.

Properties:

  • \psi(z) = -\gamma + \sum_{n=0}^\infty \frac1{n+1} - \frac1{z+1}.
  • \psi(1) = -\gamma.
  • \psi(k+1) = -\gamma +(1+\frac12+\frac13+\cdots+\frac1{k}).
  • \psi'(z) = \sum_{n=0}^\infty \frac1{(z+n)^2}.

Source: “The Functions of Mathematical Physics” by Harry Hochstadt

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