conjugate Poisson integral

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31: 4.3 Graphics

… ►Corresponding points share the same letters, with bars signifying complex conjugates. … ►

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	A	B	C	$\overline{\mathrm{C}}$	D	$\overline{\mathrm{D}}$	E	$\overline{\mathrm{E}}$	F
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Figure 4.3.2: Conformal mapping of exponential and logarithm. … Magnify

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32: 18.20 Hahn Class: Explicit Representations

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18.20.3 $$w(x;a,b,\overline{a},\overline{b})p_n(x;a,b,\overline{a},\overline{b})=\frac{1}{n!}{\delta }_x^n\left(w(x;a+\frac{1}{2}n,b+\frac{1}{2}n,\overline{a}+\frac{1}{2}n,\overline{b}+\frac{1}{2}n)\right).$$

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Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i)

Permalink:

http://dlmf.nist.gov/18.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

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18.20.9 $$p_n(x;a,b,\overline{a},\overline{b})=\frac{{\mathrm{i}}^n{\left(a+\overline{a}\right)}_n{\left(a+\overline{b}\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+2\mathrm{\Re }\left(a+b\right)-1,a+\mathrm{i}x}{a+\overline{a},a+\overline{b}};1).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19, §18.20(ii), §18.21(ii), §18.22(iii), §18.26(ii)

Permalink:

http://dlmf.nist.gov/18.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(ii), §18.20 and Ch.18

►(For symmetry properties of $p_n(x;a,b,\overline{a},\overline{b})$ with respect to $a$, $b$, $\overline{a}$, $\overline{b}$ see Andrews et al. (1999, Corollary 3.3.4).) …

33: 18.22 Hahn Class: Recurrence Relations and Differences

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18.22.4 $$q_n(x)=p_n(x;a,b,\overline{a},\overline{b})/p_n(\mathrm{i}a;a,b,\overline{a},\overline{b}),$$

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Symbols:

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

Referenced by:

§18.22(i)

Permalink:

http://dlmf.nist.gov/18.22.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

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18.22.13 $$p_n(x)=p_n(x;a,b,\overline{a},\overline{b}),$$

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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:

§18.22(ii)

Permalink:

http://dlmf.nist.gov/18.22.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

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$A(x)=(x+\mathrm{i}\overline{a})(x+\mathrm{i}\overline{b}),$

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18.22.27 $${\delta }_x\left(p_n(x;a,b,\overline{a},\overline{b})\right)=(n+2\mathrm{\Re }\left(a+b\right)-1)p_{n-1}(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2}),$$

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Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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18.22.28 $${\delta }_x\left(w(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})p_n(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})\right)=-(n+1)w(x;a,b,\overline{a},\overline{b})p_{n+1}(x;a,b,\overline{a},\overline{b}).$$

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Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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34: 19.22 Quadratic Transformations

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§19.22(i) Complete Integrals

… ►If or , then $p_{+}$ and $p_{-}$ are complex conjugates. … ►

§19.22(iii) Incomplete Integrals

… ►If or , then $z_{+}$ and $z_{-}$ are complex conjugates. However, if $x$ and $y$ are complex conjugates and $z$ and $p$ are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and $p_{\pm }^2-p_{\mp }^2=\pm |p^2-x^2|\ne 0$. …

35: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

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§1.17(ii) Integral Representations

►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): …The inner integral does not converge. … ►

Sine and Cosine Functions

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36: Bibliography J

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

Encodings:

BibTeX

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Algorithm 73: Incomplete elliptic integrals. Comm. ACM 4 (12), pp. 543.

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Notes:

Includes Algol program. For certification see same journal 6 (1963), p. 167, for remark see same journal 5 (1962), p. 514

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(iii).

Encodings:

BibTeX

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Asymptotic behavior of integrals. SIAM Rev. 14 (2), pp. 286–317.

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External Links:

Document, MathReview (A. Erdelyi), ZentralBlatt

Referenced by:

Ch.2.

Encodings:

BibTeX

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

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37: 13.20 Uniform Asymptotic Approximations for Large $\mu$

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13.20.17 $$M_{\kappa ,\mu }\left(x\right)={\left(8\mu \right)}^{\frac{1}{4}}{\left(\frac{2\mu }{\mathrm{e}}\right)}^{2\mu }{\left(\frac{\mathrm{e}}{\kappa +\mu }\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\left(U(\mu -\kappa ,-\zeta \sqrt{2\mu })+\mathrm{env}\overline{U}(\mu -\kappa ,\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$ , $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function

Referenced by:

§13.20(iv), §13.20(iv)

Permalink:

http://dlmf.nist.gov/13.20.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

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13.20.18 $$W_{\kappa ,\mu }\left(x\right)={\left(\frac{1}{2}\mu \right)}^{-\frac{1}{4}}{\left(\frac{\kappa +\mu }{\mathrm{e}}\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\left(U(\mu -\kappa ,\zeta \sqrt{2\mu })+\mathrm{env}\overline{U}(\mu -\kappa ,-\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$ , $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function

Referenced by:

§13.20(iv), §13.20(iv)

Permalink:

http://dlmf.nist.gov/13.20.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

… ►For the parabolic cylinder functions $U$ and $\overline{U}$ see §12.2, and for the $\mathrm{env}$ functions associated with $U$ and $\overline{U}$ see §14.15(v). …

38: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
11.16(iii) Integrals of …														✓					✓		✓	✓	a
…
11.16(vi) Integrals of …														✓							✓	✓	a
…
12.21(ii) $U(a,x)$, $V(a,x)$, $\overline{U}(a,x)$, $W(a,x)$, $x,a\in ℝ$						✓						a	✓				✓		✓		✓	✓	a
12.21(iii) $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, $W(a,z)$, $z,a\in ℂ$												a	✓								✓	✓	a
…
19 Elliptic Integrals
…

…

39: 36.12 Uniform Approximation of Integrals

§36.12 Uniform Approximation of Integrals

… ►The canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals. …If , then we may evaluate the complex conjugate of $I$ for real values of $𝐲$ and $g$, and obtain $I$ by conjugation and analytic continuation. … ►For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. … ►For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).

40: 12.21 Software

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