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21: 8.17 Incomplete Beta Functions

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8.17.1 $${\mathrm{B}}_x(a,b)={\int }_0^x{t}^{a-1}{(1-t)}^{b-1}dt,$$

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Defines:

${\mathrm{B}}_x(a,b)$: incomplete beta function

Symbols:

$dx$: differential of $x$ , $\int$: integral, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.1

Permalink:

http://dlmf.nist.gov/8.17.E1

Encodings:

pMML, png, TeX

See also:

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

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§8.17(iii) Integral Representation

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8.17.10 $$I_x(a,b)=\frac{x^a{(1-x)}^b}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{s}^{-a}{(1-s)}^{-b}\frac{ds}{s-x},$$

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Symbols:

$I_x(a,b)$: incomplete beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $a$: parameter, $b$: parameter, $x$: variable and $c$: parameter

Permalink:

http://dlmf.nist.gov/8.17.E10

Encodings:

pMML, png, TeX

See also:

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

►where and the branches of $s^{-a}$ and ${(1-s)}^{-b}$ are continuous on the path and assume their principal values when $s=c$. ►Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6. …

22: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►(obtained from (5.2.1) by rotation of the integration path) is also needed. ►

§8.21(iii) Integral Representations

… ►In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin. … ►

23: 36.2 Catastrophes and Canonical Integrals

§36.2 Catastrophes and Canonical Integrals

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§36.2(i) Definitions

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Canonical Integrals

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§36.2(iii) Symmetries

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24: Errata

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The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$. The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at $s=a$ for $\gamma (a,z)$. In the case of (8.6.12), it separates the poles of the gamma function from the poles at $s=0,1,2,\mathrm{\dots }$.

Reported 2017-07-10 by Kurt Fischer.

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25: 11.5 Integral Representations

§11.5 Integral Representations

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§11.5(i) Integrals Along the Real Line

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§11.5(ii) Contour Integrals

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Mellin–Barnes Integrals

… ►In (11.5.8) and (11.5.9) the path of integration separates the poles of the integrand at $s=0,1,2,\mathrm{\dots }$ from those at $s=-1,-2,-3,\mathrm{\dots }$. …

26: 5.12 Beta Function

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Euler’s Beta Integral

… ►In (5.12.11) and (5.12.12) the fractional powers are continuous on the integration paths and take their principal values at the beginning. … ►

Pochhammer’s Integral

►When $a,b\in ℂ$ … ►

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Figure 5.12.3: $t$-plane. Contour for Pochhammer’s integral. Magnify

27: 15.6 Integral Representations

§15.6 Integral Representations

►The function $𝐅(a,b;c;z)$ (not $F(a,b;c;z)$) has the following integral representations: … ►In all cases the integrands are continuous functions of $t$ on the integration paths, except possibly at the endpoints. Note that (15.6.8) can be rewritten as a fractional integral. … ►

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Figure 15.6.1: $t$-plane. … Magnify

28: 10.9 Integral Representations

… ►where the integration path is a simple loop contour, and $t^{\nu +1}$ is continuous on the path and takes its principal value at the intersection with the positive real axis. … ►In (10.9.20) and (10.9.21) the integration paths are simple loop contours not enclosing $t=-1$. Also, ${(t^2-1)}^{\nu -\frac{1}{2}}$ is continuous on the path, and takes its principal value at the intersection with the interval $(1,\mathrm{\infty })$. … ►where the integration path passes to the left of $t=0,1,2,\mathrm{\dots }$. …where $c$ is a positive constant and the integration path encloses the points $t=0,-1,-2,\mathrm{\dots }$. …

29: 31.10 Integral Equations and Representations

… ►For suitable choices of the branches of the $P$-symbols in (31.10.9) and the contour $C$, we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution). …

30: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.7 $$\frac{1}{\pi }{\int }_{{ℒ}_j}{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$ $j=3,4$,

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths

A&S Ref:

20.7.18 (in slightly different notation)

Referenced by:

§28.28(i)

Permalink:

http://dlmf.nist.gov/28.28.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.8 $$\frac{1}{\pi }{\int }_{{ℒ}_j}2\mathrm{i}h\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }^{\prime }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$ $j=3,4$,

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths

A&S Ref:

20.7.19 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.28.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.9 $$\frac{1}{2\pi }{\int }_{{ℒ}_1}{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(1)}(z,h).$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths

Referenced by:

§28.28(i)

Permalink:

http://dlmf.nist.gov/28.28.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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