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31: 14.17 Integrals

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14.17.5 $${\int }_0^1{x}^{\sigma }{\left(1-x^2\right)}^{\mu /2}{𝖯}_{\nu }^{-\mu }\left(x\right)dx=\frac{\mathrm{\Gamma }\left(\frac{1}{2}\sigma +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\sigma +1\right)}{2^{\mu +1}\mathrm{\Gamma }\left(\frac{1}{2}\sigma -\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)\mathrm{\Gamma }\left(\frac{1}{2}\sigma +\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{2}\right)},$$ $\mathrm{\Re }\sigma >-1$, $\mathrm{\Re }\mu >-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(ii), §14.17 and Ch.14

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§14.17(iv) Definite Integrals of Products

►With $\psi \left(x\right)={\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ (§5.2(i)), …

32: 9.11 Products

§9.11 Products

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§9.11(i) Differential Equation

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§9.11(ii) Wronskian

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§9.11(iv) Indefinite Integrals

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§9.11(v) Definite Integrals

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33: 9.10 Integrals

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9.10.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(-t\right)dt=\frac{1}{3}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}+\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\right),$$ $\mathrm{\Re }p>0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: 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 and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

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9.10.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Bi}\left(-t\right)dt=\frac{1}{\sqrt{3}}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}-\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$ $\mathrm{\Re }p>0$.

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: 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 and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ and the incomplete gamma function $\mathrm{\Gamma }$ see §§13.1, 13.2, and 8.2(i). ►For Laplace transforms of products of Airy functions see Shawagfeh (1992). … ►

9.10.17 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\mathrm{Ai}\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\alpha \right)}{3^{(\alpha +2)/3}\mathrm{\Gamma }\left(\frac{1}{3}\alpha +\frac{2}{3}\right)},$$ $\mathrm{\Re }\alpha >0$.

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Defines:

$\alpha$: parameter (locally)

Symbols:

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

Keywords:

Mellin transform

Source:

Olver (1997b, (6.10), p. 338)

Permalink:

http://dlmf.nist.gov/9.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(vi), §9.10 and Ch.9

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34: 31.14 General Fuchsian Equation

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31.14.3 $$w(z)=\left(\prod _{j=1}^N{(z-a_j)}^{-{\gamma }_j/2}\right)W(z),$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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35: 24.14 Sums

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24.14.12 $$det[E_{r+s}]={(-1)}^{n(n+1)/2}{\left(\prod _{k=1}^{n}k!\right)}^2.$$

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Symbols:

$E_n$: Euler numbers, $det$: determinant, $!$: factorial (as in $n!$), $k$: integer, $n$: integer, $r$: row and $s$: column

Referenced by:

§24.14(ii)

Permalink:

http://dlmf.nist.gov/24.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(ii), §24.14 and Ch.24

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36: 18.28 Askey–Wilson Class

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18.28.19 $$R_n(x)=R_n(x;\alpha ,\beta ,\gamma ,\delta |q)=\sum _{\mathrm{\ell }=0}^n\frac{q^{\mathrm{\ell }}{(q^{-n},\alpha \beta q^{n+1};q)}_{\mathrm{\ell }}}{{(\alpha q,\beta \delta q,\gamma q,q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-q^{j}x+\gamma \delta q^{2j+1})={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},\alpha \beta q^{n+1},q^{-y},\gamma \delta q^{y+1}}{\alpha q,\beta \delta q,\gamma q};q,q),$$ $\alpha q$, $\beta \delta q$, or $\gamma q=q^{-N}$; $n=0,1,\mathrm{\dots },N$.

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Defines:

$R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial and $R_n(x)$ (locally)

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28 and Ch.18

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37: 18.17 Integrals

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§18.17(ii) Integral Representations for Products

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Ultraspherical

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Legendre

… ►For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively. … ►For the beta function $\mathrm{B}(a,b)$ see §5.12, and for the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. …

38: 11.7 Integrals and Sums

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11.7.11 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -\nu -1}{𝐇}_{\nu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\frac{1}{2}\mu \right)2^{\mu -\nu -1}\tan \left(\frac{1}{2}\pi \mu \right)}{\mathrm{\Gamma }\left(\nu -\frac{1}{2}\mu +1\right)},$$ , $\mathrm{\Re }\nu >\mathrm{\Re }\mu -\frac{3}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\tan z$: tangent function and $\nu$: real or complex order

Keywords:

Mellin transform

A&S Ref:

12.1.27

Permalink:

http://dlmf.nist.gov/11.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(ii), §11.7 and Ch.11

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11.7.12 $${\int }_0^{\mathrm{\infty }}{t}^{-\mu -\nu }{𝐇}_{\mu }\left(t\right){𝐇}_{\nu }\left(t\right)dt=\frac{\sqrt{\pi }\mathrm{\Gamma }\left(\mu +\nu \right)}{2^{\mu +\nu }\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)},$$ $\mathrm{\Re }\left(\mu +\nu \right)>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\nu$: real or complex order

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/11.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(ii), §11.7 and Ch.11

►For other integrals involving products of Struve functions see Zanovello (1978, 1995). For integrals involving products of ${𝐌}_{\nu }\left(t\right)$ functions, see Paris and Sy (1983, Appendix). …

39: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.31 $$\frac{2}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t\sin t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^m\mathrm{i}{\gamma }_{\nu ,m}{\mathrm{D}}_0(\nu ,\nu +2m,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\gamma }_{\nu ,m}$

Permalink:

http://dlmf.nist.gov/28.28.E31

Encodings:

pMML, png, TeX

See also:

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

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28.28.32 $$\frac{\sinh \left(2z\right)}{{\pi }^2}{\int }_0^{2\pi }\frac{{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}{\gamma }_{\nu ,m}{\mathrm{D}}_1(\nu ,\nu +2m,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\gamma }_{\nu ,m}$

Permalink:

http://dlmf.nist.gov/28.28.E32

Encodings:

pMML, png, TeX

See also:

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

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28.28.33 $${\gamma }_{\nu ,m}=\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{me}}_{\nu }^{\prime }\left(t\right){\mathrm{me}}_{-\nu -2m}\left(t\right)dt={(-1)}^m\frac{4\mathrm{i}}{\pi }\frac{{\mathrm{me}}_{\nu }^{\prime }\left(0\right){\mathrm{me}}_{-\nu -2m}\left(0\right)}{{\mathrm{D}}_1(\nu ,\nu +2m,0)}.$$

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Defines:

${\gamma }_{\nu ,m}$ (locally)

Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.28.E33

Encodings:

pMML, png, TeX

See also:

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

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28.28.44 $$\frac{1}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin \left(2t\right){\mathrm{se}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^p\mathrm{i}{\widehat{\gamma }}_{n,m}{\mathrm{Dsc}}_0(n,m,z),$$

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Symbols:

${\mathrm{Dsc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\gamma }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E44

Encodings:

pMML, png, TeX

See also:

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

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28.28.46 $${\widehat{\gamma }}_{n,m}=\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{se}}_n^{\prime }(t,h^2){\mathrm{ce}}_m(t,h^2)dt={(-1)}^{p+1}\frac{4}{\mathrm{i}\pi }\frac{{\mathrm{se}}_n^{\prime }(0,h^2){\mathrm{ce}}_m(0,h^2)}{{\mathrm{Dsc}}_1(n,m,0)}.$$

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Defines:

${\widehat{\gamma }}_{n,m}$ (locally)

Symbols:

${\mathrm{Dsc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.28.E46

Encodings:

pMML, png, TeX

See also:

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

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40: 25.15 Dirichlet $L$-functions

… ►For the principal character ${\chi }_1(modk)$, $L(s,{\chi }_1)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\varphi \left(k\right)/k$, where $\varphi \left(k\right)$ is Euler’s totient function (§27.2). … ►

25.15.2 $$L(s,\chi )=\prod _p{\left(1-\frac{\chi (p)}{p^s}\right)}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Re }$: real part, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

infinite product, product representation

Source:

Apostol (1976, p. 231)

Permalink:

http://dlmf.nist.gov/25.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

►with the product taken over all primes $p$, beginning with $p=2$. … ►

25.15.4 $$L(s,\chi )=L(s,{\chi }_0)\prod _{p|k}\left(1-\frac{{\chi }_0(p)}{p^s}\right),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

product representation

Source:

Apostol (1976, p. 262)

Referenced by:

§25.15(i)

Permalink:

http://dlmf.nist.gov/25.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

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