# improper integral

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## 7 matching pages

##### 2: 25.14 Lerch’s Transcendent
25.14.5 $\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^% {s-1}e^{-ax}}{1-ze^{-x}}\,\mathrm{d}x,$ $\Re s>1$, $\Re a>0$ if $z=1$; $\Re s>0$, $\Re a>0$ if $z\in\mathbb{C}\setminus[1,\infty)$.
25.14.6 $\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{% s}}\,\mathrm{d}x-2\int_{0}^{\infty}\frac{\sin\left(x\ln z-s\operatorname{% arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\,\mathrm{d}x,$ $\Re a>0$ if $\left|z\right|<1$; $\Re s>1$, $\Re a>0$ if $\left|z\right|=1$.
##### 3: 25.12 Polylogarithms
25.12.11 $\operatorname{Li}_{s}\left(z\right)\equiv\frac{z}{\Gamma\left(s\right)}\int_{0% }^{\infty}\frac{x^{s-1}}{e^{x}-z}\,\mathrm{d}x,$
25.12.14 $F_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x}% +1}\,\mathrm{d}t,$ $s>-1$,
25.12.15 $G_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x}% -1}\,\mathrm{d}t,$ $s>-1$, $x<0$; or $s>0$, $x\leq 0$,
##### 4: 25.11 Hurwitz Zeta Function
25.11.5 $\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1% }-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{(x+a)^{s+1}}\,\mathrm% {d}x,$ $s\neq 1$, $\Re s>0$, $a>0$, $N=0,1,2,3,\dots$.
25.11.26 $\zeta\left(s,a\right)=-s\int_{-a}^{\infty}\frac{x-\left\lfloor x\right\rfloor-% \frac{1}{2}}{(x+a)^{s+1}}\,\mathrm{d}x,$ $-1<\Re s<0$, $0.
25.11.29 $\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}% \frac{\sin\left(s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{% s/2}(e^{2\pi x}-1)}\,\mathrm{d}x,$ $s\neq 1$, $\Re a>0$.
25.11.31 $\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{2\cosh x}% \,\mathrm{d}x=4^{-s}\left(\zeta\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\zeta% \left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right),$ $\Re s>0$, $\Re a>-1$.
##### 5: 25.16 Mathematical Applications
25.16.6 $H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta\left(s\right)+\frac{1}{2}% \zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r\right)\zeta\left(s+2r% \right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\widetilde{B}% _{2k+1}\left(x\right)}{x^{2k+2}}\,\mathrm{d}x,$
25.16.7 $H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\,\mathrm{d}x.$
##### 6: 25.2 Definition and Expansions
25.2.8 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N% }^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{s+1}}\,\mathrm{d}x,$ $\Re s>0$, $N=1,2,3,\dots$.
25.2.9 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}% {2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}% N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{% \widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,$ $\Re s>-2n$; $n,N=1,2,3,\dots$.
##### 7: 28.32 Mathematical Applications
defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $\mathbb{C}$. …