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1: 8.17 Incomplete Beta Functions

… ►Throughout §§8.17 and 8.18 we assume that $a>0$, $b>0$, and $0\le x\le 1$. … ►

8.17.4 $$I_x(a,b)=1-I_{1-x}(b,a).$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.3 (Symmetry relation)

Referenced by:

§8.17(v), §8.18(i)

Permalink:

http://dlmf.nist.gov/8.17.E4

Encodings:

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See also:

Annotations for §8.17(i), §8.17 and Ch.8

… ►With $a>0$, $b>0$, and , … ►

8.17.13 $$(a+b)I_x(a,b)=a{I}_x(a+1,b)+b{I}_x(a,b+1),$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.7

Permalink:

http://dlmf.nist.gov/8.17.E13

Encodings:

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See also:

Annotations for §8.17(iv), §8.17 and Ch.8

►

8.17.14 $$(a+bx)I_x(a,b)=xb{I}_x(a-1,b+1)+a{I}_x(a+1,b),$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

Permalink:

http://dlmf.nist.gov/8.17.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(iv), §8.17 and Ch.8

…

2: 8.10 Inequalities

… ►

8.10.1 $$x^{1-a}{\mathrm{e}}^x\mathrm{\Gamma }(a,x)\le 1,$$ $x>0$, ,

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $x$: real variable and $a$: parameter

Referenced by:

§8.10

Permalink:

http://dlmf.nist.gov/8.10.E1

Encodings:

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See also:

Annotations for §8.10 and Ch.8

►

8.10.2 $$\gamma (a,x)\ge \frac{x^{a-1}}{a}(1-{\mathrm{e}}^{-x}),$$ $x>0$, .

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $x$: real variable and $a$: parameter

Referenced by:

§8.10

Permalink:

http://dlmf.nist.gov/8.10.E2

Encodings:

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See also:

Annotations for §8.10 and Ch.8

… ►

8.10.3 $$x^{1-a}{\mathrm{e}}^x\mathrm{\Gamma }(a,x)=1+\frac{a-1}{x}\vartheta ,$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter and $\vartheta$

Permalink:

http://dlmf.nist.gov/8.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.10 and Ch.8

… ►

8.10.4 $x>0$, $a\le 2$.

ⓘ

Symbols:

$x$: real variable, $a$: parameter and $\vartheta$

Referenced by:

§8.10

Permalink:

http://dlmf.nist.gov/8.10.E4

Encodings:

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Annotations for §8.10 and Ch.8

… ►

8.10.11 $${(1-{\mathrm{e}}^{-{\alpha }_{a}x})}^a\le P(a,x)\le {(1-{\mathrm{e}}^{-{\beta }_{a}x})}^a,$$ $x\ge 0$, $a>0$,

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $x$: real variable, $a$: parameter, ${\alpha }_a$ and ${\beta }_a$

Referenced by:

(7.8.8), §8.10

Permalink:

http://dlmf.nist.gov/8.10.E11

Encodings:

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See also:

Annotations for §8.10, §8.10 and Ch.8

…

3: 8.5 Confluent Hypergeometric Representations

… ►

8.5.1 $$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.12

Referenced by:

§8.11(ii), §8.14

Permalink:

http://dlmf.nist.gov/8.5.E1

Encodings:

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Annotations for §8.5 and Ch.8

►

8.5.2 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}𝐌(1,1+a,z)=𝐌(a,1+a,-z).$$

ⓘ

Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.5.E2

Encodings:

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Annotations for §8.5 and Ch.8

►

8.5.3 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.14, §8.19(vi)

Permalink:

http://dlmf.nist.gov/8.5.E3

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.4 $$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/8.5.E4

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.5 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}a-\frac{1}{2}}{W}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

4: 8.8 Recurrence Relations and Derivatives

… ►

8.8.1 $$\gamma (a+1,z)=a\gamma (a,z)-z^a{\mathrm{e}}^{-z},$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.22

Permalink:

http://dlmf.nist.gov/8.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

►

8.8.2 $$\mathrm{\Gamma }(a+1,z)=a\mathrm{\Gamma }(a,z)+z^a{\mathrm{e}}^{-z}.$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.19(v)

Permalink:

http://dlmf.nist.gov/8.8.E2

Encodings:

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See also:

Annotations for §8.8 and Ch.8

… ►

8.8.3 $$w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0.$$

ⓘ

Symbols:

$z$: complex variable, $a$: parameter and $w(a,z)$: function

Permalink:

http://dlmf.nist.gov/8.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

… ►

8.8.5 $$P(a+1,z)=P(a,z)-\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.21

Referenced by:

§8.25(v)

Permalink:

http://dlmf.nist.gov/8.8.E5

Encodings:

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Annotations for §8.8 and Ch.8

►

8.8.6 $$Q(a+1,z)=Q(a,z)+\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.25(v), §8.4

Permalink:

http://dlmf.nist.gov/8.8.E6

Encodings:

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Annotations for §8.8 and Ch.8

…

5: Bibliography U

… ►

Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.

ⓘ

External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

►

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.

ⓘ

External Links:

ISSN 0305-0041, Document, MathReview (F. I. Ibragimov), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

►

Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.

ⓘ

External Links:

ISSN 0305-0041, Document, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§14.26.

Encodings:

BibTeX

…

6: 8.21 Generalized Sine and Cosine Integrals

… ►

8.21.20 $$\mathrm{si}(a,z)=f(a,z)\cos z+g(a,z)\sin z,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $f(a,z)$: auxiliary function and $g(a,z)$: auxiliary function

Referenced by:

§8.21(viii)

Permalink:

http://dlmf.nist.gov/8.21.E20

Encodings:

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See also:

Annotations for §8.21(vii), §8.21 and Ch.8

►

8.21.21 $$\mathrm{ci}(a,z)=-f(a,z)\sin z+g(a,z)\cos z.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $f(a,z)$: auxiliary function and $g(a,z)$: auxiliary function

Referenced by:

§8.21(viii)

Permalink:

http://dlmf.nist.gov/8.21.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

… ►

8.21.22 $$f(a,z)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{{(t+z)}^{1-a}}dt,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter and $f(a,z)$: auxiliary function

A&S Ref:

5.2.12 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E22

Encodings:

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See also:

Annotations for §8.21(vii), §8.21 and Ch.8

►

8.21.23 $$g(a,z)={\int }_0^{\mathrm{\infty }}\frac{\cos t}{{(t+z)}^{1-a}}dt.$$

ⓘ

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $a$: parameter and $g(a,z)$: auxiliary function

A&S Ref:

5.2.13 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

… ►

8.21.24 $$f(a,z)=\frac{z^a}{2}{\int }_0^{\mathrm{\infty }}\left({(1+\mathrm{i}t)}^{a-1}+{(1-\mathrm{i}t)}^{a-1}\right){\mathrm{e}}^{-zt}dt,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $a$: parameter and $f(a,z)$: auxiliary function

Referenced by:

§8.21(vii), §8.21(viii)

Permalink:

http://dlmf.nist.gov/8.21.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

…

7: 31.3 Basic Solutions

… ►

31.3.2 $$a\gamma c_1-q{c}_0=0,$$

ⓘ

Symbols:

$\gamma$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter and $c_j$: coefficients

Permalink:

http://dlmf.nist.gov/31.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(i), §31.3 and Ch.31

… ►

31.3.5 $$z^{1-\gamma }H\mathrm{\ell }(a,(a\delta +ϵ)(1-\gamma )+q;\alpha +1-\gamma ,\beta +1-\gamma ,2-\gamma ,\delta ;z).$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.3(iii), §31.7(ii)

Permalink:

http://dlmf.nist.gov/31.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(i), §31.3 and Ch.31

… ►

31.3.6 $$H\mathrm{\ell }(1-a,\alpha \beta -q;\alpha ,\beta ,\delta ,\gamma ;1-z),$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E6

Encodings:

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See also:

Annotations for §31.3(ii), §31.3 and Ch.31

►

31.3.7 $${(1-z)}^{1-\delta }H\mathrm{\ell }(1-a,((1-a)\gamma +ϵ)(1-\delta )+\alpha \beta -q;\alpha +1-\delta ,\beta +1-\delta ,2-\delta ,\gamma ;1-z).$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(ii), §31.3 and Ch.31

… ►

31.3.8 $$H\mathrm{\ell }(\frac{a}{a-1},\frac{\alpha \beta a-q}{a-1};\alpha ,\beta ,ϵ,\delta ;\frac{a-z}{a-1}),$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(ii), §31.3 and Ch.31

…

8: 8.2 Definitions and Basic Properties

… ►

8.2.2 $$\mathrm{\Gamma }(a,z)={\int }_z^{\mathrm{\infty }}{t}^{a-1}{\mathrm{e}}^{-t}dt,$$

ⓘ

Defines:

$\mathrm{\Gamma }(a,z)$: incomplete gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.3

Referenced by:

(8.15.2), §8.2(i), §8.2(iii), §8.21(iii), §8.6(i)

Permalink:

http://dlmf.nist.gov/8.2.E2

Encodings:

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See also:

Annotations for §8.2(i), §8.2 and Ch.8

… ►

8.2.3 $$\gamma (a,z)+\mathrm{\Gamma }(a,z)=\mathrm{\Gamma }\left(a\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.3

Referenced by:

§8.6(i), §8.7

Permalink:

http://dlmf.nist.gov/8.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

… ►

8.2.5 $$P(a,z)+Q(a,z)=1.$$

ⓘ

Symbols:

$P(a,z)$: normalized incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

… ►

8.2.8 $$\gamma (a,z{\mathrm{e}}^{2\pi m\mathrm{i}})={\mathrm{e}}^{2\pi m\mathrm{i}a}\gamma (a,z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/8.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(ii), §8.2 and Ch.8

… ►

8.2.12 $$\frac{d^{2}w}{{dz}^2}+\left(1+\frac{1-a}{z}\right)\frac{dw}{dz}=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: parameter and $w$: solution

Permalink:

http://dlmf.nist.gov/8.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(iii), §8.2 and Ch.8

…

9: 8.7 Series Expansions

… ►

8.7.1 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}=\frac{1}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(a+k)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

A&S Ref:

6.5.29

Referenced by:

§8.11(ii), §8.14, §8.6(i), §8.7

Permalink:

http://dlmf.nist.gov/8.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

►

8.7.2 $$\gamma (a,x+y)-\gamma (a,x)=\mathrm{\Gamma }(a,x)-\mathrm{\Gamma }(a,x+y)={\mathrm{e}}^{-x}{x}^{a-1}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{{(-x)}^n}(1-{\mathrm{e}}^{-y}{e}_n(y)),$$ .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions

A&S Ref:

6.5.30

Permalink:

http://dlmf.nist.gov/8.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

►

8.7.3 $$\mathrm{\Gamma }(a,z)=\mathrm{\Gamma }\left(a\right)-\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{z}^{a+k}}{k!(a+k)}=\mathrm{\Gamma }\left(a\right)\left(1-z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.14, §8.19(iv), §8.4, §8.7

Permalink:

http://dlmf.nist.gov/8.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

►

8.7.4 $$\gamma (a,x)=\mathrm{\Gamma }\left(a\right)x^{\frac{1}{2}a}{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}{e}_n(-1)x^{\frac{1}{2}n}{I}_{n+a}\left(2x^{1/2}\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions

Permalink:

http://dlmf.nist.gov/8.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

… ►

8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

…

10: 8.14 Integrals

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8.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}\frac{\gamma (b,x)}{\mathrm{\Gamma }\left(b\right)}dx=\frac{{(1+a)}^{-b}}{a},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }b>-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Referenced by:

§8.14, §8.14

Permalink:

http://dlmf.nist.gov/8.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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8.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}\mathrm{\Gamma }(b,x)dx=\mathrm{\Gamma }\left(b\right)\frac{1-{(1+a)}^{-b}}{a},$$ $\mathrm{\Re }a>-1$, $\mathrm{\Re }b>-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

A&S Ref:

6.5.36

Referenced by:

§8.14, §8.14

Permalink:

http://dlmf.nist.gov/8.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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8.14.3 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}\gamma (b,x)dx=-\frac{\mathrm{\Gamma }\left(a+b\right)}{a},$$ , $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/8.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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8.14.4 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}\mathrm{\Gamma }(b,x)dx=\frac{\mathrm{\Gamma }\left(a+b\right)}{a},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

A&S Ref:

6.5.37

Permalink:

http://dlmf.nist.gov/8.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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8.14.5 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}{\mathrm{e}}^{-sx}\gamma (b,x)dx=\frac{\mathrm{\Gamma }\left(a+b\right)}{b{(1+s)}^{a+b}}F(1,a+b;1+b;1/(1+s)),$$ $\mathrm{\Re }s>0$, $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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