# real part

(0.010 seconds)

## 1—10 of 249 matching pages

##### 1: 27.4 Euler Products and Dirichlet Series
27.4.3 $\zeta\left(s\right)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p}(1-p^{-s})^{-1},$ $\Re s>1$.
27.4.5 $\sum_{n=1}^{\infty}\mu\left(n\right)n^{-s}=\frac{1}{\zeta\left(s\right)},$ $\Re s>1$,
27.4.11 $\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)% \zeta\left(s-\alpha\right),$ $\Re s>\max(1,1+\Re\alpha)$,
27.4.12 $\sum_{n=1}^{\infty}\Lambda\left(n\right)n^{-s}=-\frac{\zeta'\left(s\right)}{% \zeta\left(s\right)},$ $\Re s>1$,
##### 2: 8.14 Integrals
8.14.1 $\int_{0}^{\infty}e^{-ax}\frac{\gamma\left(b,x\right)}{\Gamma\left(b\right)}\,% \mathrm{d}x=\frac{(1+a)^{-b}}{a},$ $\Re a>0$, $\Re b>-1$,
8.14.2 $\int_{0}^{\infty}e^{-ax}\Gamma\left(b,x\right)\,\mathrm{d}x=\Gamma\left(b% \right)\frac{1-(1+a)^{-b}}{a},$ $\Re a>-1$, $\Re b>-1$.
8.14.3 $\int_{0}^{\infty}x^{a-1}\gamma\left(b,x\right)\,\mathrm{d}x=-\frac{\Gamma\left% (a+b\right)}{a},$ $\Re a<0$, $\Re\left(a+b\right)>0$,
8.14.4 $\int_{0}^{\infty}x^{a-1}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{\Gamma\left(% a+b\right)}{a},$ $\Re a>0$, $\Re\left(a+b\right)>0$,
8.14.6 $\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{% \Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1+a;s/(1+s)\right),$ $\Re s>-1$, $\Re\left(a+b\right)>0$, $\Re a>0$.
##### 3: 13.10 Integrals
13.10.3 $\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a,c,kt\right)\,\mathrm{d}t=% \Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;\ifrac{k}{z}% \right),$ $\Re b>0$, $\Re z>\max\left(\Re k,0\right)$,
13.10.5 $\int_{0}^{\infty}e^{-t}t^{b-1}{\mathbf{M}}\left(a,c,t\right)\,\mathrm{d}t=% \frac{\Gamma\left(b\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)% \Gamma\left(c-b\right)},$ $\Re\left(c-a\right)>\Re b>0$,
13.10.10 $\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,b,-t\right)\,\mathrm{d}t=% \frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)}{\Gamma\left(a% \right)\Gamma\left(b-\lambda\right)},$ $0<\Re\lambda<\Re a$,
13.10.11 $\int_{0}^{\infty}t^{\lambda-1}U\left(a,b,t\right)\,\mathrm{d}t=\frac{\Gamma% \left(\lambda\right)\Gamma\left(a-\lambda\right)\Gamma\left(\lambda-b+1\right)% }{\Gamma\left(a\right)\Gamma\left(a-b+1\right)},$ $\max\left(\Re b-1,0\right)<\Re\lambda<\Re a$.
13.10.13 $\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}{\mathbf{M}}\left(a,b,t\right)J_{% \nu}\left(2\sqrt{xt}\right)\,\mathrm{d}t=x^{-a+\frac{1}{2}\nu}e^{-x}{\mathbf{M% }}\left(\nu-b+1,\nu-a+1,x\right),$ $x>0$, $2\Re a<\Re\nu+\tfrac{5}{2}$, $\Re b>0$,
##### 4: 25.14 Lerch’s Transcendent
25.14.1 ${\Phi\left(z,s,a\right)\equiv\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$ $|z|<1$; $\Re s>1,|z|=1$.
25.14.2 $\zeta\left(s,a\right)=\Phi\left(1,s,a\right),$ $\Re s>1$, $a\neq 0,-1,-2,\dots$,
25.14.3 $\operatorname{Li}_{s}\left(z\right)=z\Phi\left(z,s,1\right),$ $\Re s>1$, $|z|\leq 1$.
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 $\left|z\right|=1$.
##### 5: 5.13 Integrals
5.13.1 ${\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+a\right)\Gamma\left% (b-s\right)z^{-s}\,\mathrm{d}s=\frac{\Gamma\left(a+b\right)z^{a}}{(1+z)^{a+b}}},$ $\Re\left(a+b\right)>0$, $-\Re a, $|\operatorname{ph}z|<\pi$.
5.13.3 $\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma\left(a+it\right)\Gamma\left(b+it% \right)\Gamma\left(c-it\right)\Gamma\left(d-it\right)\,\mathrm{d}t=\frac{% \Gamma\left(a+c\right)\Gamma\left(a+d\right)\Gamma\left(b+c\right)\Gamma\left(% b+d\right)}{\Gamma\left(a+b+c+d\right)},$ $\Re a,\Re b,\Re c,\Re d>0$.
5.13.4 $\int_{-\infty}^{\infty}\frac{\,\mathrm{d}t}{\Gamma\left(a+t\right)\Gamma\left(% b+t\right)\Gamma\left(c-t\right)\Gamma\left(d-t\right)}=\frac{\Gamma\left(a+b+% c+d-3\right)}{\Gamma\left(a+c-1\right)\Gamma\left(a+d-1\right)\Gamma\left(b+c-% 1\right)\Gamma\left(b+d-1\right)},$ $\Re\left(a+b+c+d\right)>3$.
5.13.5 $\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{k=1}^{4}\Gamma\left(a_{k}+it% \right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2it\right)\Gamma\left(-2it% \right)}\,\mathrm{d}t=\frac{\prod_{1\leq j $\Re\left(a_{k}\right)>0$, $k=1,2,3,4$.
##### 6: 25.13 Periodic Zeta Function
where $\Re s>1$ if $x$ is an integer, $\Re s>0$ otherwise. …
25.13.2 $F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(2\pi)^{1-s}}\*\left(e^{\pi i(% 1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}\zeta\left(1-s,1-x\right)\right),$ $0, $\Re s>1$,
25.13.3 $\zeta\left(1-s,x\right)=\frac{\Gamma\left(s\right)}{(2\pi)^{s}}\left(e^{-\pi is% /2}F\left(x,s\right)+e^{\pi is/2}F\left(-x,s\right)\right),$ $\Re s>0$ if $0; $\Re s>1$ if $x=1$.
##### 7: 7.9 Continued Fractions
7.9.1 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{z}{z^{2}+\cfrac{\frac{1}{2}}{1+% \cfrac{1}{z^{2}+\cfrac{\frac{3}{2}}{1+\cfrac{2}{z^{2}+\cdots}}}}},$ $\Re z>0$,
7.9.2 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{2z}{2z^{2}+1-\cfrac{1\cdot 2}{2% z^{2}+5-\cfrac{3\cdot 4}{2z^{2}+9-\cdots}}},$ $\Re z>0$,
##### 8: 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 $\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t% }}\right)\,\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+% \sqrt{b})},$ $|\operatorname{ph}a|<\frac{1}{2}\pi$, $\Re b>0$, $\Re c\geq 0$.
##### 9: 16.15 Integral Representations and Integrals
16.15.1 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\frac{\Gamma\left(% \gamma\right)}{\Gamma\left(\alpha\right)\Gamma\left(\gamma-\alpha\right)}\int_% {0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(1-ux)^{\beta}(1-uy)^{\beta^% {\prime}}}\,\mathrm{d}u,$ $\Re\alpha>0$, $\Re\left(\gamma-\alpha\right)>0$,
16.15.2 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=% \frac{\Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left% (\beta\right)\Gamma\left(\beta^{\prime}\right)\Gamma\left(\gamma-\beta\right)% \Gamma\left(\gamma^{\prime}-\beta^{\prime}\right)}\int_{0}^{1}\!\!\!\int_{0}^{% 1}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u)^{\gamma-\beta-1}(1-v)^{\gamma^{% \prime}-\beta^{\prime}-1}}{(1-ux-vy)^{\alpha}}\,\mathrm{d}u\,\mathrm{d}v,$ $\Re\gamma>\Re\beta>0$, $\Re\gamma^{\prime}>\Re\beta^{\prime}>0$,
16.15.3 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta\right)\Gamma\left(\beta^{% \prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime}\right)}\iint_{\Delta}% \frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-\beta-\beta^{\prime}-1}}{% (1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\,\mathrm{d}u\,\mathrm{d}v,$ $\Re\left(\gamma-\beta-\beta^{\prime}\right)>0$, $\Re\beta>0$, $\Re\beta^{\prime}>0$,
16.15.4 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\frac{% \Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-\alpha\right)\Gamma% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\,\mathrm{d}u\,\mathrm{d}v,$ $\Re\gamma>\Re\alpha>0$, $\Re\gamma^{\prime}>\Re\beta>0$.
##### 10: 15.14 Integrals
15.14.1 $\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\,\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},$ $\min(\Re a,\Re b)>\Re s>0$.