2. 令 ψ(z)=d[lnΓ(z)]dz\begin{aligned}\psi(z) = \frac{d[\ln \Gamma(z)]}{dz}\end{aligned}ψ(z)=dzd[lnΓ(z)]
① 证明ψ(z+1)=1z+ψ(z)\begin{aligned} \psi(z+1) = \frac{1}{z} + \psi(z)\end{aligned}ψ(z+1)=z1+ψ(z)
证：
ψ(z)=d[lnΓ(z)]dz=[Γ(z)]′Γ(z)\begin{aligned} \psi(z) = \frac{d[\ln \Gamma(z)]}{dz} = \frac{[\Gamma(z)]'}{\Gamma(z)} \end{aligned}ψ(z)=dzd[lnΓ(z)]=Γ(z)[Γ(z)]′
ψ(z+1)=[Γ(z+1)]′Γ(z+1)=[zΓ(z)]′z⋅Γ(z)=Γ(z)+z[Γ(z)]′zΓ(z)\begin{aligned} \psi(z+1) = \frac{[\Gamma(z+1)]'}{\Gamma(z+1)} = \frac{[z\Gamma(z)]'}{z\cdot\Gamma(z)} = \frac{\Gamma(z) + z[\Gamma(z)]'}{z\Gamma(z)} \end{aligned}ψ(z+1)=Γ(z+1)[Γ(z+1)]′=z⋅Γ(z)[zΓ(z)]′=zΓ(z)Γ(z)+z[Γ(z)]′
=1z+[Γ(z)]′Γ(z)=1z+ψ(z)\begin{aligned} = \frac{1}{z} + \frac{[\Gamma(z)]'}{\Gamma(z)} = \frac{1}{z} + \psi(z) \end{aligned}=z1+Γ(z)[Γ(z)]′=z1+ψ(z)
② 证明 ψ(1−z)=ψ(z)+πcotπz\begin{aligned}\psi(1-z) = \psi(z) + \pi \cot \pi z\end{aligned}ψ(1−z)=ψ(z)+πcotπz
证：
ψ(z)=d[lnΓ(z)]dz=[Γ(z)]′Γ(z)\begin{aligned} \psi(z) = \frac{d[\ln \Gamma(z)]}{dz} = \frac{[\Gamma(z)]'}{\Gamma(z)} \end{aligned}ψ(z)=dzd[lnΓ(z)]=Γ(z)[Γ(z)]′
ψ(1−z)=d[lnΓ(1−z)]d(1−z)=−[Γ(1−z)]′Γ(1−z)\begin{aligned} \psi(1-z) = \frac{d[\ln \Gamma(1-z)]}{d(1-z)} = -\frac{[\Gamma(1-z)]'}{\Gamma(1-z)} \end{aligned}ψ(1−z)=d(1−z)d[lnΓ(1−z)]=−Γ(1−z)[Γ(1−z)]′
=−[πsinπz⋅1Γ(z)]′⋅sinπz⋅Γ(z)π\begin{aligned} = -\left[ \frac{\pi}{\sin \pi z} \cdot \frac{1}{\Gamma(z)} \right]' \cdot \frac{\sin \pi z \cdot \Gamma(z)}{\pi} \end{aligned}=−[sinπzπ⋅Γ(z)1]′⋅πsinπz⋅Γ(z)
=−−π[sinπz⋅Γ(z)]2[πcosπz⋅Γ(z)+sinπz⋅Γ′(z)]⋅sinπz⋅Γ(z)π\begin{aligned} = -\frac{-\pi}{[\sin \pi z \cdot \Gamma(z)]^2} [\pi \cos \pi z \cdot \Gamma(z) + \sin \pi z \cdot \Gamma'(z)] \cdot \frac{\sin \pi z \cdot \Gamma(z)}{\pi} \end{aligned}=−[sinπz⋅Γ(z)]2−π[πcosπz⋅Γ(z)+sinπz⋅Γ′(z)]⋅πsinπz⋅Γ(z)
=1sinπz⋅Γ(z)[πcosπz⋅Γ(z)+sinπz⋅Γ′(z)]\begin{aligned} = \frac{1}{\sin \pi z \cdot \Gamma(z)} [\pi \cos \pi z \cdot \Gamma(z) + \sin \pi z \cdot \Gamma'(z)] \end{aligned}=sinπz⋅Γ(z)1[πcosπz⋅Γ(z)+sinπz⋅Γ′(z)]
=πcotπz+Γ′(z)Γ(z)\begin{aligned} = \pi \cot \pi z + \frac{\Gamma'(z)}{\Gamma(z)} \end{aligned}=πcotπz+Γ(z)Γ′(z)
=πcotπz+ψ(z)\begin{aligned} = \pi \cot \pi z + \psi(z) \end{aligned}=πcotπz+ψ(z)