transformations of parameters

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11: 9.10 Integrals

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9.10.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(t\right)dt={\mathrm{e}}^{-p^3/3}\left(\frac{1}{3}-\frac{p{}_1{}^{}F_1^{}(\frac{1}{3};\frac{4}{3};\frac{1}{3}{p}^3)}{3^{4/3}\mathrm{\Gamma }\left(\frac{4}{3}\right)}+\frac{p^2{}_1{}^{}F_1^{}(\frac{2}{3};\frac{5}{3};\frac{1}{3}{p}^3)}{3^{5/3}\mathrm{\Gamma }\left(\frac{5}{3}\right)}\right),$$ $p\in ℂ$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E14

Encodings:

pMML, png, TeX

See also:

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

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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

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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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12: Bibliography T

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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13: Bibliography L

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Asymptotics of the first Appell function $F_1$ with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

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Asymptotics of the first Appell function $F_1$ with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

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14: 2.8 Differential Equations with a Parameter

… ►In Case III $f(z)$ has a simple pole at $z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z_0$. …

15: 19.15 Advantages of Symmetry

… ►(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …

16: 32.7 Bäcklund Transformations

… ►With the exception of ${\text{P}}_{\text{I}}$, a Bäcklund transformation relates a Painlevé transcendent of one type either to another of the same type but with different values of the parameters, or to another type. … ►and ►

32.7.44 $${\theta }_j={\mathrm{\Theta }}_j+\frac{1}{2}\sigma ,$$

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Symbols:

${\mathrm{\Theta }}_j$: transformation, ${\theta }_j$: parameters and $\sigma$

Permalink:

http://dlmf.nist.gov/32.7.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(vii), §32.7 and Ch.32

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32.7.45 $$\sigma ={\theta }_0+{\theta }_1+{\theta }_2+{\theta }_{\mathrm{\infty }}-1=1-({\mathrm{\Theta }}_0+{\mathrm{\Theta }}_1+{\mathrm{\Theta }}_2+{\mathrm{\Theta }}_{\mathrm{\infty }}).$$

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Symbols:

${\mathrm{\Theta }}_j$: transformation, ${\theta }_j$: parameters and $\sigma$

Permalink:

http://dlmf.nist.gov/32.7.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(vii), §32.7 and Ch.32

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32.7.46 $$\frac{\sigma }{w-W}=\frac{z(z-1)W^{\prime }}{W(W-1)(W-z)}+\frac{{\mathrm{\Theta }}_0}{W}+\frac{{\mathrm{\Theta }}_1}{W-1}+\frac{{\mathrm{\Theta }}_2-1}{W-z}=\frac{z(z-1)w^{\prime }}{w(w-1)(w-z)}+\frac{{\theta }_0}{w}+\frac{{\theta }_1}{w-1}+\frac{{\theta }_2-1}{w-z}.$$

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Symbols:

$z$: real, ${\mathrm{\Theta }}_j$: transformation, ${\theta }_j$: parameters, $\sigma$ and $W(\zeta ,\epsilon /2)$: transformation

Permalink:

http://dlmf.nist.gov/32.7.E46

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(vii), §32.7 and Ch.32

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17: 18.25 Wilson Class: Definitions

… ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. ►

Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

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OP	$p_n(x)$	$x=\lambda (y)$	Orthogonality range for $y$	Constraints
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18: 2.4 Contour Integrals

… ►Then …

19: 1.14 Integral Transforms

… ►If $x^{\sigma -1}f(x)$ is integrable on $(0,\mathrm{\infty })$ for all $\sigma$ in , then the integral (1.14.32) converges and $ℳf\left(s\right)$ is an analytic function of $s$ in the vertical strip . … ►

1.14.33 $$\underset{t\to \pm \mathrm{\infty }}{lim}ℳf\left(\sigma +\mathrm{i}t\right)=0.$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\mathrm{i}$: imaginary unit and $\sigma \in (a,b)$: parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/1.14.E33

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §1.14(iv), §1.14 and Ch.1

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1.14.35 $$f(x)=\frac{1}{2\pi \mathrm{i}}{\int }_{\sigma -\mathrm{i}\mathrm{\infty }}^{\sigma +\mathrm{i}\mathrm{\infty }}{x}^{-s}ℳf\left(s\right)ds.$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\sigma \in (a,b)$: parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/1.14.E35

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

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1.14.37 $${\int }_0^{\mathrm{\infty }}f(x)g(x)dx=\frac{1}{2\pi \mathrm{i}}{\int }_{\sigma -\mathrm{i}\mathrm{\infty }}^{\sigma +\mathrm{i}\mathrm{\infty }}ℳf\left(1-s\right)ℳg\left(s\right)ds.$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\sigma \in (a,b)$: parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/1.14.E37

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

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1.14.49 $$\underset{t\to 0+}{lim}\frac{𝒮f\left(-\sigma -\mathrm{i}t\right)-𝒮f\left(-\sigma +\mathrm{i}t\right)}{2\pi \mathrm{i}}=\frac{1}{2}(f(\sigma +)+f(\sigma -)),$$

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Symbols:

$𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\sigma \in (a,b)$: parameter

Permalink:

http://dlmf.nist.gov/1.14.E49

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Stieltjes transform was changed to $𝒮f\left(s\right)$ from $𝒮(f;s)$.

See also:

Annotations for §1.14(vi), §1.14(vi), §1.14 and Ch.1

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20: 15.8 Transformations of Variable

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15.8.12 $$𝐅(a,b;a+b-m;z)={(1-z)}^{-m}𝐅(\tilde{a},\tilde{b};\tilde{a}+\tilde{b}+m;z),$$ $\tilde{a}=a-m,\tilde{b}=b-m$.

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/15.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

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15.8.28 $$\frac{2\sqrt{z}\mathrm{\Gamma }\left(-\frac{1}{2}\right)\mathrm{\Gamma }\left(a+b-\frac{1}{2}\right)}{\mathrm{\Gamma }\left(a-\frac{1}{2}\right)\mathrm{\Gamma }\left(b-\frac{1}{2}\right)}F(a,b;\frac{3}{2};z)=F(2a-1,2b-1;a+b-\frac{1}{2};\frac{1}{2}-\frac{1}{2}\sqrt{z})-F(2a-1,2b-1;a+b-\frac{1}{2};\frac{1}{2}+\frac{1}{2}\sqrt{z}),$$ , .

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Referenced by:

§15.4(i)

Permalink:

http://dlmf.nist.gov/15.8.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

… ►When the intersection of two groups in Table 15.8.1 is not empty there exist special quadratic transformations, with only one free parameter, between two hypergeometric functions in the same group. …