Fonksiyona dîgamayê
Fonksiyona dîgamayê an jî vatiniya dîgamayê, li gor termînolojiya matematîkê wek muşteqeke logarîtmîk a fonksiyona gamayê dihê pênasekirin. Ya yekem a ji fonksiyonên polîgamayê ye. ψ ( x ) = d d x ln ⁡ Γ ( x ) = Γ ′ ( x ) Γ ( x ) {\displaystyle \psi (x)={\frac {d}{dx}}\ln {\Gamma (x)}={\frac {\Gamma '(x)}{\Gamma (x)}}} ψ ( n ) = H n − 1 − γ {\displaystyle \psi (n)=H_{n-1}-\gamma } , H n {\displaystyle H_{n}} - n {\displaystyle n} , γ {\displaystyle \gamma } ψ ( 1 − x ) − ψ ( x ) = π cot ⁡ ( π x ) {\displaystyle \psi (1-x)-\psi (x)=\pi \cot(\pi x)} ψ ( x + 1 ) = ψ ( x ) + 1 x {\displaystyle \psi (x+1)=\psi (x)+{\frac {1}{x}}} ψ ( x ) = ln ⁡ ( x ) − 1 2 x + ∑ n = 1 ∞ ζ ( 1 − 2 n ) x 2 n {\displaystyle \psi (x)=\ln(x)-{\frac {1}{2x}}+\sum _{n=1}^{\infty }{\frac {\zeta (1-2n)}{x^{2n}}}} ζ ( x ) {\displaystyle \zeta (x)} ψ ( x ) = ∑ n = 0 ∞ 1 n + 1 ∑ k = 0 n ( − 1 ) k ( n k ) ln ⁡ ( x + k ) {\displaystyle \psi (x)=\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}\ln(x+k)} Γ ′ ( p / q ) Γ ( p / q ) = − γ − ln ⁡ ( 2 q ) − π 2 cot ⁡ ( π p q ) + 2 ∑ 0 < n < q / 2 cos ⁡ ( 2 π p n q ) ln ⁡ ( sin ⁡ ( π n q ) ) {\displaystyle {\frac {\Gamma '(p/q)}{\Gamma (p/q)}}=-\gamma -\ln(2q)-{\frac {\pi }{2}}\cot \left({\frac {\pi p}{q}}\right)+2\sum _{0<n<q/2}\cos \left({\frac {2\pi pn}{q}}\right)\ln \left(\sin \left({\frac {\pi n}{q}}\right)\right)} , p , q {\displaystyle p,q} , 0 < p < q {\displaystyle 0<p<q} .