Chapter 11.11, Problem 7P

The function $\psi (p)=\frac{d}{dp}\ln \Gamma (p)$ is called the digamma function, and the polygamma functions are defined by ${\psi }^n(p)=\frac{d^n}{d{p}^n}\psi (p).$ [Warning: Some authors define $\psi (p)$ as

(a) Show that $\psi (p+1)=\psi (p)+\frac{1}{p}.$ Hint: See (3.4).

(b) Use Problem 6 to obtain $\psi (p)~\ln p-\frac{1}{2p}-\frac{1}{12p^2}\cdots$.