User:Eric Kvaalen/Notes/Digamma

(From, by User:HenningThielemann)

Some proofs for properties of the digamma function.

Recurrence relation


\begin{align} \ln(\Gamma(x+1)) &= \ln(\Gamma(x)\cdot x) &&= \ln(\Gamma(x)) + \ln x & \left| \ \frac{\mathrm{d}}{\mathrm{d} x} \right. \\ \psi(x+1) & &&= \psi(x) + \frac{1}{x} \end{align} $$

Bounds
We apply the mean-value theorem to the function $$\ln \circ\, \Gamma$$ on the positive real axis and get:



\begin{align} \exists \xi\in(x,x+1)\ (\ln \circ \Gamma)'(\xi) &= \frac{\ln(\Gamma(x+1)) - \ln(\Gamma(x))}{1} \\ \psi(\xi) &= \ln(\Gamma(x)\cdot x) - \ln(\Gamma(x))\\ &= \ln x \\ \psi(\xi) &< \ln \xi \qquad \text{(because of monotonicity of logarithm)} \\ \psi(x+1) &> \ln x \qquad \text{(because of monotonicity of digamma)} \end{align} $$

Asymptotic series for exp o digamma
In this section we give the concrete computation for the coefficients of the two asymptotic series for $$\exp \circ\, \psi$$. The functions asymptoticSeries1 and asymptoticSeries2 compute the coefficients, whereas the functions asymptotic1 and asymptotic2 compute approximations, given the number of terms and the digamma argument. The program is self-contained Haskell 98 code.