# scaled gamma function

(0.004 seconds)

## 1—10 of 36 matching pages

##### 1: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$
###### General Case
Let $\widetilde{\Gamma}(z)$ denote the scaled gamma function
8.18.13 $\widetilde{\Gamma}(z)=(2\pi)^{-1/2}e^{z}z^{(1/2)-z}\Gamma\left(z\right),$
8.18.14 $I_{x}\left(a,b\right)\sim Q\left(b,a\zeta\right)-\frac{(2\pi b)^{-1/2}}{% \widetilde{\Gamma}(b)}\left(\frac{x}{x_{0}}\right)^{a}\left(\frac{1-x}{1-x_{0}% }\right)^{b}\sum_{k=0}^{\infty}\frac{h_{k}(\zeta,\mu)}{a^{k}},$
##### 2: 14.3 Definitions and Hypergeometric Representations
14.3.4 $\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{2% ^{m}\Gamma\left(\nu-m+1\right)}\left(1-x^{2}\right)^{m/2}\mathbf{F}\left(\nu+m% +1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right);$
14.3.5 $\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{% \Gamma\left(\nu-m+1\right)}\left(\frac{1-x}{1+x}\right)^{m/2}\mathbf{F}\left(% \nu+1,-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right).$
14.3.13 $w_{1}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{% 1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)}\left(1-x^{2}% \right)^{-\mu/2}\mathbf{F}\left(-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}% \nu-\tfrac{1}{2}\mu+\tfrac{1}{2};\tfrac{1}{2};x^{2}\right),$
14.3.14 $w_{2}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1% \right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}x\left(1% -x^{2}\right)^{-\mu/2}\mathbf{F}\left(\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2% }\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2};x^{2}\right).$
14.3.19 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{2^{\nu}\Gamma\left(\nu+1\right)% (x+1)^{\mu/2}}{(x-1)^{(\mu/2)+\nu+1}}\mathbf{F}\left(\nu+1,\nu+\mu+1;2\nu+2;% \frac{2}{1-x}\right),$
##### 3: 5.23 Approximations
See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\Gamma\left(z\right)$. …
##### 4: 15.14 Integrals
15.14.1 $\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},$ $\min(\Re a,\Re b)>\Re s>0$.
##### 5: 16.2 Definition and Analytic Properties
16.2.5 ${{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b};z\right)=\ifrac{{{}_{p}F_{% q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)}{\left(\Gamma% \left(b_{1}\right)\cdots\Gamma\left(b_{q}\right)\right)}=\sum_{k=0}^{\infty}% \frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{\Gamma\left(b_{1% }+k\right)\cdots\Gamma\left(b_{q}+k\right)}\frac{z^{k}}{k!};$
##### 6: 14.23 Values on the Cut
14.23.3 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=\frac{e^{\mp\nu\pi i/2}\pi^{3/2% }\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\frac{x\mathbf{F}\left(\frac{1}% {2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};% x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)% \Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}\mp i\frac{% \mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+% \frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}% \mu+1\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}\right).$
##### 8: 14.1 Special Notation
 $x$, $y$, $\tau$ real variables. … Olver’s scaled hypergeometric function: $\ifrac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}$.
##### 9: 10.22 Integrals
10.22.49 $\int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}\left(bt\right)\mathrm{d}t=\frac{(% \tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\Gamma\left(\mu+\nu\right)\*\mathbf{F}\left(% \frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),$ $\Re\left(\mu+\nu\right)>0,\Re\left(a\pm ib\right)>0$,
10.22.50 $\int_{0}^{\infty}t^{\mu-1}e^{-at}Y_{\nu}\left(bt\right)\mathrm{d}t=\cot\left(% \nu\pi\right)\frac{(\tfrac{1}{2}b)^{\nu}\Gamma\left(\mu+\nu\right)}{(a^{2}+b^{% 2})^{\frac{1}{2}(\mu+\nu)}}\*\mathbf{F}\left(\frac{\mu+\nu}{2},\frac{1-\mu+\nu% }{2};\nu+1;\frac{b^{2}}{a^{2}+b^{2}}\right)-\csc\left(\nu\pi\right)\frac{(% \tfrac{1}{2}b)^{-\nu}\Gamma\left(\mu-\nu\right)}{(a^{2}+b^{2})^{\frac{1}{2}(% \mu-\nu)}}\*\mathbf{F}\left(\frac{\mu-\nu}{2},\frac{1-\mu-\nu}{2};1-\nu;\frac{% b^{2}}{a^{2}+b^{2}}\right),$ $\Re\mu>|\Re\nu|,\Re\left(a\pm ib\right)>0$.
10.22.56 $\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{\nu}\left(bt\right)}{t^{% \lambda}}\mathrm{d}t=\frac{a^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu-% \frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}b^{\mu-\lambda+1}\Gamma\left% (\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\lambda+\frac{1}{2}\right)}\*\mathbf% {F}\left(\tfrac{1}{2}(\mu+\nu-\lambda+1),\tfrac{1}{2}(\mu-\nu-\lambda+1);\mu+1% ;\frac{a^{2}}{b^{2}}\right),$ $0, $\Re\left(\mu+\nu+1\right)>\Re\lambda>-1$.
10.22.58 $\int_{0}^{\infty}\frac{J_{\nu}\left(at\right)J_{\nu}\left(bt\right)}{t^{% \lambda}}\mathrm{d}t=\frac{(ab)^{\nu}\Gamma\left(\nu-\frac{1}{2}\lambda+\frac{% 1}{2}\right)}{2^{\lambda}(a^{2}+b^{2})^{\nu-\frac{1}{2}\lambda+\frac{1}{2}}% \Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\right)}\mathbf{F}\left(\frac{2\nu+1% -\lambda}{4},\frac{2\nu+3-\lambda}{4};\nu+1;\frac{4a^{2}b^{2}}{(a^{2}+b^{2})^{% 2}}\right),$ $a\neq b$, $\Re\left(2\nu+1\right)>\Re\lambda>-1$.
10.22.64 $\int_{0}^{\infty}J_{\mu+2n+1}\left(at\right)J_{\mu}\left(bt\right)\mathrm{d}t=% \begin{cases}\dfrac{b^{\mu}\Gamma\left(\mu+n+1\right)}{a^{\mu+1}n!}\mathbf{F}% \left(-n,\mu+n+1;\mu+1;\dfrac{b^{2}}{a^{2}}\right),&00),\\ 0,&0