F-homotopic transformations

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11: 16.6 Transformations of Variable

§16.6 Transformations of Variable

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Quadratic

… ►

Cubic

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16.6.2

{{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

►For Kummer-type transformations of

{{}_{2}F_{2}}

functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).

12: 17.18 Methods of Computation

§17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …

13: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …

14: 19.13 Integrals of Elliptic Integrals

… ►

§19.13(iii) Laplace Transforms

►For direct and inverse Laplace transforms for the complete elliptic integrals

K\left(k\right)

E\left(k\right)

, and

D\left(k\right)

see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.

15: 3.9 Acceleration of Convergence

… ►

§3.9(i) Sequence Transformations

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§3.9(ii) Euler’s Transformation of Series

… ►Euler’s transformation is usually applied to alternating series. … ►

§3.9(iv) Shanks’ Transformation

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§3.9(vi) Applications and Further Transformations

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16: 32.7 Bäcklund Transformations

§32.7 Bäcklund Transformations

… ►Then the transformations … ►The quadratic transformation …The quartic transformation … ►

17: 2.6 Distributional Methods

… ►

§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of

f(t)

is defined by … ►

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)

being the Mellin transform of

f(t)

or its analytic continuation (§2.5(ii)). … ►Corresponding results for the generalized Stieltjes transform … ►where

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)

is the Mellin transform of

f

or its analytic continuation. …

18: 7.14 Integrals

… ►

Fourier Transform

… ►

Laplace Transforms

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7.14.2

\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\,\mathrm{d}t=\frac{1% }{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),

\Re a>0

|\operatorname{ph}b|<\tfrac{1}{4}\pi

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.17
Referenced by:: §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

►

7.14.3

\int_{0}^{\infty}e^{-at}\operatorname{erf}\sqrt{bt}\,\mathrm{d}t=\frac{1}{a}% \sqrt{\frac{b}{a+b}},

\Re a>0

\Re b>0

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.19
Referenced by:: §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

… ►

Laplace Transforms

…

19: Vadim B. Kuznetsov

… ►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …

20: 1.16 Distributions

… ►

§1.16(vii) Fourier Transforms of Tempered Distributions

… ►Then its Fourier transform is … ►If

u\in\mathcal{T}^{*}_{n}

is a tempered distribution, then its Fourier transform

\mathscr{F}\left(u\right)

is defined by …The Fourier transform

\mathscr{F}\left(u\right)

of a tempered distribution is again a tempered distribution, and … ►

§1.16(viii) Fourier Transforms of Special Distributions

…