# Olver hypergeometric function

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##### 2: 15.6 Integral Representations
15.6.1 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\Re c>\Re b>0$.
15.6.2 $\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\,% \mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $c-b\neq 1,2,3,\dots$, $\Re b>0$.
15.6.6 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)% \Gamma\left(b\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma% \left(a+t\right)\Gamma\left(b+t\right)\Gamma\left(-t\right)}{\Gamma\left(c+t% \right)}(-z)^{t}\,\mathrm{d}t,$ $|\operatorname{ph}\left(-z\right)|<\pi$; $a,b\neq 0,-1,-2,\dots$.
15.6.7 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)% \Gamma\left(b\right)\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\Gamma\left(a+t\right)\Gamma\left(b+t% \right)\Gamma\left(c-a-b-t\right)\Gamma\left(-t\right)(1-z)^{t}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $a,b,c-a,c-b\neq 0,-1,-2,\dots$.
15.6.8 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\Re c>\Re d>0$.
##### 3: 13.1 Special Notation
The main functions treated in this chapter are the Kummer functions $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$, Olver’s function ${\mathbf{M}}\left(a,b,z\right)$, and the Whittaker functions $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$. …
##### 4: 14.3 Definitions and Hypergeometric Representations
14.3.3 $\mathbf{F}\left(a,b;c;x\right)=\frac{1}{\Gamma\left(c\right)}F\left(a,b;c;x\right)$
is Olver’s hypergeometric function15.1). …
14.3.6 $P^{\mu}_{\nu}\left(x\right)=\left(\frac{x+1}{x-1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$
14.3.9 $P^{-\mu}_{\nu}\left(x\right)=\left(\frac{x-1}{x+1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$
14.3.15 $P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}% \left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$
##### 5: 15.7 Continued Fractions
15.7.1 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},$
15.7.3 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},$
15.7.5 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},$
##### 6: 15.15 Sums
15.15.1 $\mathbf{F}\left({a,b\atop c};\frac{1}{z}\right)=\left(1-\frac{z_{0}}{z}\right)% ^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{s!}\*\mathbf{F}\left({-s,b% \atop c};\frac{1}{z_{0}}\right)\left(1-\frac{z}{z_{0}}\right)^{-s}.$
##### 7: 15.14 Integrals
15.14.1 $\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\,\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},$ $\min(\Re a,\Re b)>\Re s>0$.
##### 8: 15.2 Definitions and Analytical Properties
15.2.2 $\mathbf{F}\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{% \left(b\right)_{s}}}{\Gamma\left(c+s\right)s!}z^{s},$ $|z|<1$,
15.2.3 $\mathbf{F}\left({a,b\atop c};x+\mathrm{i}0\right)-\mathbf{F}\left({a,b\atop c}% ;x-\mathrm{i}0\right)=\frac{2\pi\mathrm{i}}{\Gamma\left(a\right)\Gamma\left(b% \right)}(x-1)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-x\right),$ $x>1$.
15.2.3_5 $\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}% \left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1% )!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right),$ $n=0,1,2,\dots$.
##### 9: 13.10 Integrals
13.10.1 $\int{\mathbf{M}}\left(a,b,z\right)\,\mathrm{d}z=\frac{1}{a-1}{\mathbf{M}}\left% (a-1,b-1,z\right),$
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 $\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,b,-t\right)\,\mathrm{d}t=% \frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)}{\Gamma\left(a% \right)\Gamma\left(b-\lambda\right)},$ $0<\Re\lambda<\Re a$,
13.10.14 $\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{M}}\left(a,b,t\right)J_{\nu}% \left(2\sqrt{xt}\right)\,\mathrm{d}t=\frac{x^{\frac{1}{2}\nu}e^{-x}}{\Gamma% \left(b-a\right)}U\left(a,a-b+\nu+2,x\right),$ $x>0$, $-1<\Re\nu<2\Re\left(b-a\right)-\tfrac{1}{2}$,
##### 10: 13.4 Integral Representations
13.4.2 ${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(b-c\right)}\int_{0}^{1}{% \mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-1}\,\mathrm{d}t,$ $\Re b>\Re c>0$,
13.4.3 ${\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\Gamma% \left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}J_{b-1}% \left(2\sqrt{zt}\right)\,\mathrm{d}t,$ $\Re a>0$.
13.4.12 ${\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b\right)}{2\pi\mathrm{i}}z^{1% -b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;% \ifrac{1}{t}\right)\,\mathrm{d}t,$ $b\neq 0,-1,-2,\dots$, $\left|\operatorname{ph}z\right|<\frac{1}{2}\pi$.
13.4.13 ${\mathbf{M}}\left(a,b,z\right)=\frac{z^{1-b}}{2\pi\mathrm{i}}\int_{-\infty}^{(% 0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-a}\,\mathrm{d}t,$ $|\operatorname{ph}{z}|<\frac{1}{2}\pi$.