ABSTRACT
More formulas can be obtained from the corresponding sections due to the relations Γ z = lim w → ∞ w z z 1 F 1 z ; z + 1 ; - w , Γ 1 - z Γ 1 + z = z π sin z π , Γ ( z + 1 2 ) Γ ( 1 2 - z ) = π cos z π , ψ z = z - 1 3 F 2 1 , 1 , 2 - z ; 2 , 2 ; 1 - C , ψ - z = 1 z + π cot z π + ψ z , ψ ( n ) z = - 1 n + 1 n ! z - n - 1 n + 2 F n + 1 1 , z , z , … , z ; z + 1 , z + 1 , … , z + 1 ; 1 , ψ ( n ) z ± m = ψ ( n ) z ± - 1 n n ! ∑ k = ( 1 ∓ 1 ) / 2 m - ( 1 ± 1 ) / 2 1 z ± k n + 1 , ζ s = Li s 1 , Re s > 1 ; ζ s , a + n = ζ s , a - ∑ k = 0 n - 1 1 ( a + k 2 ) s / 2 , ζ s , a - n = ζ s , a + ∑ k = 0 n - 1 1 ( a + k - n 2 ) s / 2 . $$ \begin{aligned} \Gamma \left(z\right)=\lim _{w\rightarrow \infty }\frac{w^z}{z}{\,}_1F_1\left(z;{\,}z+1;\,-w\right),\\ \Gamma \left(1-z\right)\Gamma \left(1+z\right)=\frac{z\pi }{\sin \left(z\pi \right)},\quad \Gamma \biggl (z+\frac{1}{2}\biggr )\Gamma \biggl (\frac{1}{2}-z\biggr )=\frac{\pi }{\cos \left(z\pi \right)},\\ \psi \left(z\right)=\left(z-1\right){\,}_3F_2\left(1,{\,}1,{\,}2-z;{\,}2,{\,}2;{\,}1\right)-\mathbf C ,\quad \psi \left(-z\right)=\frac{1}{z}+\pi \cot \left(z\pi \right)+\psi \left(z\right),\\ \psi ^{(n)}\left(z\right)=\left(-1\right)^{n+1} n!{\,} z^{-n-1} {\,}_{n+2}F_{n+1}\left(1,{\,}z,{\,}z,\ldots ,z;{\,}z+1,{\,}z+1,\ldots ,z+1;{\,}1\right),\\ \psi ^{(n)}\left(z\pm m\right)=\psi ^{(n)}\left(z\right)\pm \left(-1\right)^{n}n!\sum _{k=(1\mp 1)/2}^{m-(1\pm 1)/2}\frac{1}{\left(z\pm k\right)^{n+1}},\\ \zeta \left(s\right)=\text{ Li}_s\left(1\right),\quad \text{ Re}s>1;\quad \zeta \left(s,{\,}a+n\right)=\zeta \left(s,{\,}a\right)-\sum _{k=0}^{n-1}\frac{1}{\bigl (\left(a+k\right)^2\bigr )^{s/2}},\\ \zeta \left(s,{\,}a-n\right)=\zeta \left(s,{\,}a\right)+\sum _{k=0}^{n-1}\frac{1}{\bigl (\left(a+k-n\right)^2\bigr )^{s/2}}. \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429434259/189889df-fb75-4d89-b3dd-985795c10b9e/content/um11.tif"/>
