Sears’ balanced 4ϕ3 transformation

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21: Bibliography O

… ►

F. Oberhettinger and T. P. Higgins (1961) Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.

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Referenced by:: §1.14(viii).
Encodings:: BibTeX

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F. Oberhettinger (1990) Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).
Encodings:: BibTeX

… ►

F. Oberhettinger (1972) Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.

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Notes:: Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.
External Links:: ISBN 0-387-05997-0, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(v), §10.43(v), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §13.10(v), §13.23(iii), §15.14, §18.17(ix), §8.14, §8.6(iii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1970) A paradox in asymptotics. SIAM J. Math. Anal. 1 (4), pp. 533–534.

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External Links:: Document, MathReview (L. Hoischen), ZentralBlatt
Referenced by:: §2.10(iv).
Encodings:: BibTeX

… ►

F. W. J. Olver (1997a) Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.

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External Links:: ISSN 1073-2772, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §2.7(ii).
Encodings:: BibTeX

…

22: 32.2 Differential Equations

… ►They are distinct modulo Möbius (bilinear) transformations … ►

z\frac{\mathrm{d}f_{2}}{\mathrm{d}z}=f_{2}f_{4}(f_{3}-f_{1})+(\tfrac{1}{2}-\mu% _{4})f_{2}+\mu_{2}f_{4},

… ►

z\frac{\mathrm{d}f_{4}}{\mathrm{d}z}=f_{4}f_{2}(f_{1}-f_{3})+(\tfrac{1}{2}-\mu% _{2})f_{4}+\mu_{4}f_{2},

►where

\mu_{1}

\mu_{2}

\mu_{3}

\mu_{4}

are constants,

f_{1}

f_{2}

f_{3}

f_{4}

are functions of

z

, with … ►

\beta=\tfrac{1}{4}b,

…

23: 31.2 Differential Equations

… ►

$F$ -Homotopic Transformations

… ►By composing these three steps, there result

2^{3}=8

possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). ►

Homographic Transformations

… ►

Composite Transformations

►There are

8\cdot 24=192

automorphisms of equation (31.2.1) by compositions of

F

-homotopic and homographic transformations. …

24: 19.22 Quadratic Transformations

§19.22 Quadratic Transformations

… ►

Bartky’s Transformation

… ►Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … ►

25: 7.7 Integral Representations

… ►

7.7.7

\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\,\mathrm{d}t=\frac{\sqrt{\pi}}{% 4a}\left(e^{2ab}\operatorname{erfc}\left(ax+(b/x)\right)+e^{-2ab}\operatorname% {erfc}\left(ax-(b/x)\right)\right),

x>0

|\operatorname{ph}a|<\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 $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase and $x$ : real variable
A&S Ref:: 7.4.33 (in different form)
Referenced by:: §7.7(i), §7.8
Permalink:: http://dlmf.nist.gov/7.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §7.7(i), §7.7 and Ch.7

►

7.7.8

\int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\,\mathrm{d}t=\frac{\sqrt{\pi}}{% 2a}e^{-2ab},

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

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

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\operatorname{ph}$ : phase
A&S Ref:: 7.4.3 (in different form)
Referenced by:: §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §7.7(i), §7.7 and Ch.7

… ►

7.7.13

\mathrm{f}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i% \infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+\tfrac{1}{2}\right)\*\Gamma% \left(s+\tfrac{3}{4}\right)\Gamma\left(\tfrac{1}{4}-s\right)\,\mathrm{d}s,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable, $\zeta$ : change of variable and $c$ : constant
Keywords:: Mellin transform
Referenced by:: §7.7(ii), §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.7.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

►

7.7.14

\mathrm{g}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i% \infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+\tfrac{1}{2}\right)\*\Gamma% \left(s+\tfrac{1}{4}\right)\Gamma\left(\tfrac{3}{4}-s\right)\,\mathrm{d}s.

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable, $\zeta$ : change of variable and $c$ : constant
Keywords:: Mellin transform
Referenced by:: §7.7(ii), §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.7.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

►In (7.7.13) and (7.7.14) the integration paths are straight lines,

\zeta=\frac{1}{16}\pi^{2}z^{4}

, and

c

is a constant such that

0<c<\frac{1}{4}

in (7.7.13), and

0<c<\frac{3}{4}

in (7.7.14). …

26: 22.6 Elementary Identities

… ►

22.6.5

\operatorname{sn}\left(2z,k\right)=\frac{2\operatorname{sn}\left(z,k\right)% \operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)}{1-k^{2}{% \operatorname{sn}}^{4}\left(z,k\right)},

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.18.1
Permalink:: http://dlmf.nist.gov/22.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

►

22.6.6

\operatorname{cn}\left(2z,k\right)=\frac{{\operatorname{cn}}^{2}\left(z,k% \right)-{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}\left(z,% k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}=\frac{{\operatorname% {cn}}^{4}\left(z,k\right)-{k^{\prime}}^{2}{\operatorname{sn}}^{4}\left(z,k% \right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)},

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.18.2
Referenced by:: §22.6(ii)
Permalink:: http://dlmf.nist.gov/22.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

►

22.6.7

\operatorname{dn}\left(2z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z,k% \right)-k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}% \left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}=\frac{{% \operatorname{dn}}^{4}\left(z,k\right)+k^{2}{k^{\prime}}^{2}{\operatorname{sn}% }^{4}\left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}.

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.18.3
Referenced by:: §22.6(ii), Erratum (V1.0.7) for Equation (22.6.7)
Permalink:: http://dlmf.nist.gov/22.6.E7
Encodings:: pMML, png, TeX
Errata (effective with 1.0.7):: Originally the term $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}\left(z,k\right)$ was given incorrectly as $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}\left(z,k\right)$ .
Reported 2014-02-28 by Svante Janson
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

… ►

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

►

Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.

► ►►

$\operatorname{sn}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sc}\left(z,k^{\prime}\right)$	$\operatorname{dc}\left(iz,k\right)$ $\;=\;$	$\operatorname{dn}\left(z,k^{\prime}\right)$
…

► …

27: Bibliography P

… ►

J. B. Parkinson (1969) Optical properties of layer antiferromagnets with

\mathrm{K}_{2}\mathrm{NiF}_{4}

structure. J. Phys. C: Solid State Physics 2 (11), pp. 2012–2021.

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External Links:: Document
Referenced by:: §19.35(ii).
Encodings:: BibTeX

… ►

E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

… ►

A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt
Referenced by:: §1.14(vii).
Encodings:: BibTeX

… ►

W. H. Press and S. A. Teukolsky (1990) Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.

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Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992a) Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

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Notes:: Table erratum: Math. Comp. v. 66 (1997), no. 220, p. 1766.
External Links:: ISBN 2-88124-837-3, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §19.13(iii), §20.10(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.14, §9.10(ix).
Encodings:: BibTeX

…

28: 16.15 Integral Representations and Integrals

… ►

16.15.1

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\frac{\Gamma\left(% \gamma\right)}{\Gamma\left(\alpha\right)\Gamma\left(\gamma-\alpha\right)}\int_% {0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(1-ux)^{\beta}(1-uy)^{\beta^% {\prime}}}\,\mathrm{d}u,

\Re\alpha>0

\Re\left(\gamma-\alpha\right)>0

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/16.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.15 and Ch.16

►

16.15.2

{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=% \frac{\Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left% (\beta\right)\Gamma\left(\beta^{\prime}\right)\Gamma\left(\gamma-\beta\right)% \Gamma\left(\gamma^{\prime}-\beta^{\prime}\right)}\int_{0}^{1}\!\!\!\int_{0}^{% 1}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u)^{\gamma-\beta-1}(1-v)^{\gamma^{% \prime}-\beta^{\prime}-1}}{(1-ux-vy)^{\alpha}}\,\mathrm{d}u\,\mathrm{d}v,

\Re\gamma>\Re\beta>0

\Re\gamma^{\prime}>\Re\beta^{\prime}>0

ⓘ

Symbols:: ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/16.15.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.15 and Ch.16

… ►

16.15.4

{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\frac{% \Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-\alpha\right)\Gamma% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\,\mathrm{d}u\,\mathrm{d}v,

\Re\gamma>\Re\alpha>0

\Re\gamma^{\prime}>\Re\beta>0

ⓘ

Symbols:: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Source:: Burchnall and Chaundy (1940, (68))
Referenced by:: §16.15
Permalink:: http://dlmf.nist.gov/16.15.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.15 and Ch.16

… ►For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).

29: Bibliography N

… ►

D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

►

D. Naylor (1990) On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

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External Links:: ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt
Referenced by:: §10.43(v).
Encodings:: BibTeX

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D. Naylor (1996) On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.

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External Links:: ISSN 1073-2772, FullText, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §10.43(v).
Encodings:: BibTeX

… ►

G. Nemes (2015b) On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:: ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt
Referenced by:: §11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).
Encodings:: BibTeX

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E. Neuman (2004) Inequalities involving Bessel functions of the first kind. JIPAM. J. Inequal. Pure Appl. Math. 5 (4), pp. Article 94, 4 pp. (electronic).

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External Links:: ISSN 1443-5756, FullText, MathReview, ZentralBlatt
Referenced by:: §10.14.
Encodings:: BibTeX

…

30: 15.8 Transformations of Variable

§15.8 Transformations of Variable

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§15.8(i) Linear Transformations

… ►

§15.8(iii) Quadratic Transformations

… ►

§15.8(v) Cubic Transformations

… ►

Ramanujan’s Cubic Transformation

…