(1)

\begin{align*} \psi(1-z) & =\frac{dz}{d\left(1-z\right)}\frac{d}{dz}\log\Gamma\left(1-z\right)\\ & =-\frac{d}{dz}\log\frac{\pi\sin^{-1}\left(\pi z\right)}{\Gamma\left(z\right)}\\ & =-\frac{d}{dz}\left\{ \log\pi-\log\sin\left(\pi z\right)-\log\Gamma\left(z\right)\right\} \\ & =\frac{d}{dz}\log\sin\left(\pi z\right)+\frac{d}{dz}\log\Gamma\left(z\right)\\ & =\pi\tan^{-1}\left(\pi z\right)+\psi\left(z\right) \end{align*}
これより、
\[ \psi\left(1-z\right)-\psi\left(z\right)=\pi\tan^{-1}\left(\pi z\right) \]

(1)-2

ガンマ関数の相反公式
\[ \Gamma\left(z\right)\Gamma\left(1-z\right)=\pi\sin^{-1}\left(\pi z\right) \]
の両辺を\(z\)で微分すると、
\[ \Gamma\left(z\right)\psi\left(z\right)\Gamma\left(1-z\right)-\Gamma\left(z\right)\Gamma\left(1-z\right)\psi\left(1-z\right)=-\pi^{2}\sin^{-2}\left(\pi z\right)\cos\left(\pi z\right) \]
整理すると、
\[ \left(\psi\left(1-z\right)-\psi\left(z\right)\right)\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi^{2}\sin^{-1}\left(\pi z\right)\tan^{-1}\left(\pi z\right) \]
\(\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi\sin^{-1}\left(\pi z\right)\)なので、
\[ \psi\left(1-z\right)-\psi\left(z\right)=\pi\tan^{-1}\left(\pi z\right) \]
となる。

(2)

\begin{align*} \psi^{\left(n\right)}\left(1-z\right) & =\left(\frac{d}{d\left(1-z\right)}\right)^{n}\psi\left(1-z\right)\\ & =\left(-1\right)^{n}\frac{d^{n}}{dz^{n}}\psi\left(1-z\right)\\ & =\left(-1\right)^{n}\frac{d^{n}}{dz^{n}}\left(\pi\tan^{-1}\left(\pi z\right)+\psi\left(z\right)\right)\\ & =\left(-1\right)^{n}\left\{ \psi^{\left(n\right)}\left(z\right)+\pi\frac{d^{n}}{dz^{n}}\tan^{-1}\left(\pi z\right)\right\} \end{align*}
これより、
\[ \left(-1\right)^{n}\psi^{\left(n\right)}\left(1-z\right)-\psi^{\left(n\right)}\left(z\right)=\pi\frac{d^{n}}{dz^{n}}\tan^{-1}\left(\pi z\right) \]

(3)

\(n\in\mathbb{N}\)のとき、
\begin{align*} \lim_{z\rightarrow n}\frac{\psi\left(1-z\right))}{\Gamma\left(1-z\right))} & =\lim_{z\rightarrow n}\frac{\Gamma\left(z\right)}{\pi\sin^{-1}\left(\pi z\right)}\left\{ \psi\left(z\right)+\pi\tan^{-1}\left(\pi z\right)\right\} \\ & =\lim_{z\rightarrow n}\frac{\Gamma\left(z\right)\sin\left(\pi z\right)}{\pi}\left\{ H_{n-1}-\gamma+\pi\frac{\cos\left(\pi z\right)}{\sin\left(\pi z\right)}\right\} \\ & =\cos\left(\pi n\right)\Gamma\left(n\right)\\ & =\left(-1\right)^{n}\left(n-1\right)! \end{align*}