conjugate Poisson integral

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21: 12.12 Integrals

§12.12 Integrals

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Nicholson-type Integral

… ►For further integrals see §§13.10, 13.23, and use (12.7.14). … ► ►See also Barr (1968) and Lowdon (1970).

22: 12.14 The Function $W(a,x)$

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§12.14(vi) Integral Representations

►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument $z$ and parameter $a$. … ►

12.14.34 $$W(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{l(\mu )}{{(t^2+1)}^{\frac{1}{4}}}\left(\cos \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{u}}_{2s}(t)}{{(t^2+1)}^{3s}{\mu }^{4s}}-\sin \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{u}}_{2s+1}(t)}{{(t^2+1)}^{3s+\frac{3}{2}}{\mu }^{4s+2}}\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${\overline{u}}_s(t)$: solution, $l(\mu )$ and $\overline{\sigma }$

Permalink:

http://dlmf.nist.gov/12.14.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

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12.14.36 $$\overline{\sigma }={\mu }^2\overline{\xi }+\frac{1}{4}\pi ,$$

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Defines:

$\overline{\sigma }$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\overline{\xi }$

Permalink:

http://dlmf.nist.gov/12.14.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

►and $\overline{\xi }$ and the coefficients ${\overline{u}}_s(t)$ and ${\overline{v}}_s(t)$ as in §12.10(v). …

23: 14.30 Spherical and Spheroidal Harmonics

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14.30.6 $$Y_{l,-m}(\theta ,\varphi )={(-1)}^m\overline{Y_{l,m}(\theta ,\varphi )}.$$

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Symbols:

$\overline{z}$: complex conjugate, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E6

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the spherical harmonic on the right-hand side has been explicity marked up as a complex conjugate.

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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14.30.8 $${\int }_0^{2\pi }{\int }_0^{\pi }\overline{Y_{l_1,m_1}(\theta ,\varphi )}{Y}_{l_2,m_2}(\theta ,\varphi )\sin \theta d\theta d\varphi ={\delta }_{l_1,l_2}{\delta }_{m_1,m_2}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E8

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complex conjugate.

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

►See also (34.3.22), and for further related integrals see Askey et al. (1986). … ►

14.30.9 $${𝖯}_l\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \left({\varphi }_1-{\varphi }_2\right)\right)=\frac{4\pi }{2l+1}\sum _{m=-l}^l\overline{Y_{l,m}({\theta }_1,{\varphi }_1)}{Y}_{l,m}({\theta }_2,{\varphi }_2).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§18.18(ii)

Permalink:

http://dlmf.nist.gov/14.30.E9

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the first spherical harmonic in the sum over $m$ has been explicity marked up as a complex conjugate.

See also:

Annotations for §14.30(iii), §14.30(iii), §14.30 and Ch.14

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24: 1.1 Special Notation

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$x,y$	real variables.
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$\overline{𝐀}$	complex conjugate of the matrix $𝐀$
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${𝐀}^{\mathrm{H}}$	Hermitian conjugate of the matrix $𝐀$
…

►In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{\ast }$, $a$ being a complex number or a matrix; the Hermitian conjugate of $𝐀$ is usually being denoted ${𝐀}^{†}$.

25: 28.12 Definitions and Basic Properties

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28.12.5 $${\int }_0^{\pi }{\mathrm{me}}_{\nu }(x,q){\mathrm{me}}_{\nu }(-x,q)dx=\pi .$$

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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 $x$ , $\int$: integral, $q=h^2$: parameter, $x$: real variable and $\nu$: complex parameter

Referenced by:

§28.14

Permalink:

http://dlmf.nist.gov/28.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

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28.12.7 $${\int }_0^{\pi }{\mathrm{me}}_{\nu +2m}(x,q){\mathrm{me}}_{\nu +2n}(-x,q)dx=0,$$ $m\ne n$.

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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 $x$ , $\int$: integral, $m$: integer, $q=h^2$: parameter, $n$: integer, $x$: real variable and $\nu$: complex parameter

A&S Ref:

20.5.1 (in different form)

Permalink:

http://dlmf.nist.gov/28.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

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28.12.10 $$\overline{{\mathrm{me}}_{\nu }(z,q)}={\mathrm{me}}_{\overline{\nu }}(-\overline{z},\overline{q}).$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\overline{z}$: complex conjugate, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.12(ii)

Permalink:

http://dlmf.nist.gov/28.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

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26: 20.11 Generalizations and Analogs

… ►This is the discrete analog of the Poisson identity (§1.8(iv)). … ►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). …

27: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►Here bars do not denote complex conjugates; instead ►

12.10.26 $$\overline{\xi }=\frac{1}{2}t\sqrt{t^2+1}+\frac{1}{2}\ln \left(t+\sqrt{t^2+1}\right),$$

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Defines:

$\overline{\xi }$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/12.10.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(v), §12.10 and Ch.12

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12.10.27 $${\overline{u}}_s(t)={\mathrm{i}}^s{u}_s(-\mathrm{i}t),$$

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Defines:

${\overline{u}}_s(t)$: solution (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $s$: nonnegative integer and $u_s(t)$: solution

Permalink:

http://dlmf.nist.gov/12.10.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(v), §12.10 and Ch.12

►and the function $\overline{g}(\mu )$ has the asymptotic expansion … ►

12.10.30 $${\overline{v}}_s(t)={\mathrm{i}}^s{v}_s(-\mathrm{i}t).$$

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Defines:

${\overline{v}}_s(t)$: solution (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $s$: nonnegative integer and $v_s(t)$: solution

Permalink:

http://dlmf.nist.gov/12.10.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(v), §12.10 and Ch.12

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28: 18.37 Classical OP’s in Two or More Variables

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18.37.2 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $\mathrm{i}$: imaginary unit, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.3 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}{c}_j{z}^{m-j}{\overline{z}}^{n-j},$$

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Symbols:

$\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.4 $j=1,2,\mathrm{\dots },\min (m,n)$;

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Symbols:

$dx$: differential of $x$ , $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $\mathrm{i}$: imaginary unit, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.6 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}\frac{{(-1)}^j{\left(\alpha +1\right)}_{m+n-j}{\left(-m\right)}_j{\left(-n\right)}_j}{{\left(\alpha +1\right)}_m{\left(\alpha +1\right)}_{n}j!}{z}^{m-j}{\overline{z}}^{n-j}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $!$: factorial (as in $n!$), $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.8 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$: triangle polynomial, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

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29: Guide to Searching the DLMF

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Table 1: Query Examples

► ►►►►

Query	Matching records contain
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int sin	the integral of the sin function
int_$^$ sin	any definite integral of sin
…

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Table 2: Wildcard Examples

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Query	What it stands for
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co$te	Coordinate, conjugate, correlate,…
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int_$^$ sin	any definite integral of sin.

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… ►

Table 4: Font and Accent Examples

► ►►►

Query	Matches
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U bar	$\overline{U}$
…

►

…

30: 18.19 Hahn Class: Definitions

… ► ►

18.19.1 $$p_n(x)=p_n(x;a,b,\overline{a},\overline{b}),$$

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Symbols:

$\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19

Permalink:

http://dlmf.nist.gov/18.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

►

18.19.2 $$w(z;a,b,\overline{a},\overline{b})=\mathrm{\Gamma }\left(a+iz\right)\mathrm{\Gamma }\left(b+iz\right)\mathrm{\Gamma }\left(\overline{a}-iz\right)\mathrm{\Gamma }\left(\overline{b}-iz\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\overline{z}$: complex conjugate, $w(x)$: weight function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/18.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

►

18.19.3 $$w(x)=w(x;a,b,\overline{a},\overline{b})={|\mathrm{\Gamma }\left(a+\mathrm{i}x\right)\mathrm{\Gamma }\left(b+\mathrm{i}x\right)|}^2,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\overline{z}$: complex conjugate, $\mathrm{i}$: imaginary unit, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

►

18.19.4 $$h_n=\frac{2\pi \mathrm{\Gamma }\left(n+a+\overline{a}\right)\mathrm{\Gamma }\left(n+b+\overline{b}\right){|\mathrm{\Gamma }\left(n+a+\overline{b}\right)|}^2}{\left(2n+2\mathrm{\Re }\left(a+b\right)-1\right)\mathrm{\Gamma }\left(n+2\mathrm{\Re }\left(a+b\right)-1\right)n!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $h_n$

Referenced by:

§18.19

Permalink:

http://dlmf.nist.gov/18.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

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