# integration by parts

(0.003 seconds)

## 1—10 of 74 matching pages

##### 1: 7.14 Integrals
7.14.2 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\,\mathrm{d}t=\frac{1% }{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),$ $\Re a>0$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$,
7.14.3 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\sqrt{bt}\,\mathrm{d}t=\frac{1}{a}% \sqrt{\frac{b}{a+b}},$ $\Re a>0$, $\Re b>0$,
7.14.4 $\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t% }}\right)\,\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+% \sqrt{b})},$ $|\operatorname{ph}a|<\frac{1}{2}\pi$, $\Re b>0$, $\Re c\geq 0$.
##### 2: 8.15 Sums
8.15.2 $a\sum_{k=1}^{\infty}\left(\frac{{\mathrm{e}}^{2\pi\mathrm{i}k(z+h)}}{\left(2% \pi\mathrm{i}k\right)^{a+1}}\Gamma\left(a,2\pi\mathrm{i}kz\right)+\frac{{% \mathrm{e}}^{-2\pi\mathrm{i}k(z+h)}}{\left(-2\pi\mathrm{i}k\right)^{a+1}}% \Gamma\left(a,-2\pi\mathrm{i}kz\right)\right)=\zeta\left(-a,z+h\right)+\frac{z% ^{a+1}}{a+1}+\left(h-\tfrac{1}{2}\right)z^{a},$ $h\in[0,1]$.
##### 4: 6.16 Mathematical Applications
By integration by parts
##### 6: 18.17 Integrals
18.17.34 $\int_{0}^{\infty}e^{-xz}L^{(\alpha)}_{n}\left(x\right)e^{-x}x^{\alpha}\,% \mathrm{d}x=\frac{\Gamma\left(\alpha+n+1\right)z^{n}}{n!(z+1)^{\alpha+n+1}},$ $\Re z>-1$.
18.17.36 $\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)\,% \mathrm{d}x=\frac{2^{\beta+z}\Gamma\left(z\right)\Gamma\left(1+\beta+n\right){% \left(1+\alpha-z\right)_{n}}}{n!\Gamma\left(1+\beta+z+n\right)},$ $\Re z>0$.
18.17.38 $\int_{0}^{1}P_{2n}\left(x\right)x^{z-1}\,\mathrm{d}x=\frac{(-1)^{n}{\left(% \frac{1}{2}-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}z\right)_{n+1}}},$ $\Re z>0$,
18.17.39 $\int_{0}^{1}P_{2n+1}\left(x\right)x^{z-1}\,\mathrm{d}x=\frac{(-1)^{n}{\left(1-% \frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}+\frac{1}{2}z\right)_{n+1}}},$ $\Re z>-1$.
18.17.40 $\int_{0}^{\infty}e^{-ax}L^{(\alpha)}_{n}\left(bx\right)x^{z-1}\,\mathrm{d}x=% \frac{\Gamma\left(z+n\right)}{n!}\*{(a-b)^{n}}a^{-n-z}\*{{}_{2}F_{1}}\left({-n% ,1+\alpha-z\atop 1-n-z};\frac{a}{a-b}\right),$ $\Re a>0$, $\Re z>0$.
##### 7: 9.12 Scorer Functions
9.12.31 $\int_{0}^{z}\operatorname{Hi}\left(-t\right)\,\mathrm{d}t\sim\frac{1}{\pi}\ln z% +\frac{2\gamma+\ln 3}{3\pi}+\frac{1}{\pi}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{(3% k-1)!}{k!(3z^{3})^{k}},$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta$,
##### 9: 2.4 Contour Integrals
Then by integration by parts the integral … If, in addition, the corresponding integrals with $Q$ and $F$ replaced by their derivatives $Q^{(j)}$ and $F^{(j)}$, $j=1,2,\dots,m$, converge uniformly, then by repeated integrations by partsCases in which $p^{\prime}(t_{0})\neq 0$ are usually handled by deforming the integration path in such a way that the minimum of $\Re\left(zp(t)\right)$ is attained at a saddle point or at an endpoint. …
##### 10: 25.2 Definition and Expansions
25.2.9 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}% {2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}% N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{% \widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,$ $\Re s>-2n$; $n,N=1,2,3,\dots$.