# with a parameter

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##### 1: 8.17 Incomplete Beta Functions
Throughout §§8.17 and 8.18 we assume that $a>0$, $b>0$, and $0\leq x\leq 1$. …
8.17.4 $I_{x}\left(a,b\right)=1-I_{1-x}\left(b,a\right).$
With $a>0$, $b>0$, and $0, …
8.17.13 $(a+b)I_{x}\left(a,b\right)=aI_{x}\left(a+1,b\right)+bI_{x}\left(a,b+1\right),$
##### 2: 8.10 Inequalities
8.10.1 $x^{1-a}e^{x}\Gamma\left(a,x\right)\leq 1,$ $x>0$, $0,
8.10.2 $\gamma\left(a,x\right)\geq\frac{x^{a-1}}{a}(1-e^{-x}),$ $x>0$, $0.
8.10.4 $0<\vartheta\leq 1,$ $x>0$, $a\leq 2$.
8.10.11 $(1-e^{-\alpha_{a}x})^{a}\leq P\left(a,x\right)\leq(1-e^{-\beta_{a}x})^{a},$ $x\geq 0$, $a>0$,
##### 3: 8.5 Confluent Hypergeometric Representations
8.5.1 $\gamma\left(a,z\right)=a^{-1}z^{a}e^{-z}M\left(1,1+a,z\right)=a^{-1}z^{a}M% \left(a,1+a,-z\right),$ $a\neq 0,-1,-2,\dots$.
8.5.2 $\gamma^{*}\left(a,z\right)=e^{-z}{\mathbf{M}}\left(1,1+a,z\right)={\mathbf{M}}% \left(a,1+a,-z\right).$
8.5.3 $\Gamma\left(a,z\right)=e^{-z}U\left(1-a,1-a,z\right)=z^{a}e^{-z}U\left(1,1+a,z% \right).$
8.5.4 $\gamma\left(a,z\right)=a^{-1}z^{\frac{1}{2}a-\frac{1}{2}}e^{-\frac{1}{2}z}M_{% \frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$
8.5.5 $\Gamma\left(a,z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}a-\frac{1}{2}}W_{\frac{1% }{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$
##### 4: 8.8 Recurrence Relations and Derivatives
8.8.3 $w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0.$
##### 5: Bibliography U
• F. Ursell (1972) Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.
• F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.
• F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
• ##### 6: 8.21 Generalized Sine and Cosine Integrals
8.21.20 $\operatorname{si}\left(a,z\right)=f(a,z)\cos z+g(a,z)\sin z,$
8.21.21 $\operatorname{ci}\left(a,z\right)=-f(a,z)\sin z+g(a,z)\cos z.$
8.21.24 $f(a,z)=\frac{z^{a}}{2}\int_{0}^{\infty}\left((1+it)^{a-1}+(1-it)^{a-1}\right)e% ^{-zt}\,\mathrm{d}t,$
##### 7: 31.3 Basic Solutions
31.3.5 $z^{1-\gamma}\mathit{H\!\ell}\left(a,(a\delta+\epsilon)(1-\gamma)+q;\alpha+1-% \gamma,\beta+1-\gamma,2-\gamma,\delta;z\right).$
31.3.6 $\mathit{H\!\ell}\left(1-a,\alpha\beta-q;\alpha,\beta,\delta,\gamma;1-z\right),$
31.3.7 $(1-z)^{1-\delta}\*\mathit{H\!\ell}\left(1-a,((1-a)\gamma+\epsilon)(1-\delta)+% \alpha\beta-q;\alpha+1-\delta,\beta+1-\delta,2-\delta,\gamma;1-z\right).$
31.3.8 $\mathit{H\!\ell}\left(\frac{a}{a-1},\frac{\alpha\beta a-q}{a-1};\alpha,\beta,% \epsilon,\delta;\frac{a-z}{a-1}\right),$
##### 8: 8.2 Definitions and Basic Properties
8.2.2 $\Gamma\left(a,z\right)=\int_{z}^{\infty}t^{a-1}e^{-t}\,\mathrm{d}t,$
8.2.5 $P\left(a,z\right)+Q\left(a,z\right)=1.$
8.2.8 $\gamma\left(a,ze^{2\pi mi}\right)=e^{2\pi mia}\gamma\left(a,z\right),$ $a\neq 0,-1,-2,\dots$,
8.2.12 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(1+\frac{1-a}{z}\right)\frac{% \mathrm{d}w}{\mathrm{d}z}=0.$
##### 9: 8.7 Series Expansions
8.7.1 $\gamma^{*}\left(a,z\right)=e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(a% +k+1\right)}=\frac{1}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{% k!(a+k)}.$
8.7.2 $\gamma\left(a,x+y\right)-\gamma\left(a,x\right)=\Gamma\left(a,x\right)-\Gamma% \left(a,x+y\right)=e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{{\left(1-a\right)_{n}% }}{(-x)^{n}}(1-e^{-y}e_{n}(y)),$ $|y|<|x|$.
8.7.3 $\Gamma\left(a,z\right)=\Gamma\left(a\right)-\sum_{k=0}^{\infty}\frac{(-1)^{k}z% ^{a+k}}{k!(a+k)}=\Gamma\left(a\right)\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}% \frac{z^{k}}{\Gamma\left(a+k+1\right)}\right),$ $a\neq 0,-1,-2,\dots$.
8.7.4 $\gamma\left(a,x\right)=\Gamma\left(a\right)x^{\frac{1}{2}a}e^{-x}\sum_{n=0}^{% \infty}e_{n}(-1)x^{\frac{1}{2}n}I_{n+a}\left(\textstyle 2x^{1/2}\right),$ $a\neq 0,-1,-2,\dots$.
8.7.6 $\Gamma\left(a,x\right)=x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{L^{(a)}_{n}\left(x% \right)}{n+1},$ $x>0$, $\Re a<\frac{1}{2}$.
##### 10: 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 $\int_{0}^{\infty}x^{a-1}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{\Gamma\left(% a+b\right)}{a},$ $\Re a>0$, $\Re\left(a+b\right)>0$,
8.14.5 $\int_{0}^{\infty}x^{a-1}e^{-sx}\gamma\left(b,x\right)\,\mathrm{d}x=\frac{% \Gamma\left(a+b\right)}{b(1+s)^{a+b}}\*F\left(1,a+b;1+b;1/(1+s)\right),$ $\Re s>0$, $\Re\left(a+b\right)>0$,