fractional transformations

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21: Bibliography W

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Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York.

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Notes:

Reprinted by Chelsea Publishing, New York, 1973 and AMS Chelsea Publishing, Providence, RI, 2000.

External Links:

ISBN 0-8218-2106-7, MathReview (R. C. Buck), ZentralBlatt

Referenced by:

§3.10(iii), §4.25, §4.9(i), §4.9(ii), Ch.4, §5.10.

Encodings:

BibTeX

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The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:

ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (A. G. Ramm), ZentralBlatt

Referenced by:

(9.10.13), §9.10(ix), §9.10(v).

Encodings:

BibTeX

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The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.

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External Links:

FullText, MathReview (J. D. Tamarkin), ZentralBlatt

Referenced by:

§1.14(vi), §1.15(viii).

Encodings:

BibTeX

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22: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

BibTeX

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

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External Links:

FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

Encodings:

BibTeX

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:

Problem proposed by P. Smith.

External Links:

Document

Referenced by:

(9.10.14), (9.10.15), (9.10.16), §9.10(v).

Encodings:

BibTeX

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Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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External Links:

ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

Encodings:

BibTeX

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23: 15.6 Integral Representations

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15.6.1 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^1\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ 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 $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

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15.6.2 $$𝐅(a,b;c;z)=\frac{\mathrm{\Gamma }\left(1+b-c\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(b\right)}{\int }_0^{(1+)}\frac{t^{b-1}{(t-1)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $c-b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }b>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ 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 $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.6

Permalink:

http://dlmf.nist.gov/15.6.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

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15.6.2_5 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^{\mathrm{\infty }}\frac{t^{b-1}{\left(t+1\right)}^{a-c}}{{\left(t-zt+1\right)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ 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 $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.6, Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/15.6.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.5):

This equation was added.

See also:

Annotations for §15.6 and Ch.15

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15.6.8 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(c-d\right)}{\int }_0^1𝐅(a,b;d;zt)t^{d-1}{(1-t)}^{c-d-1}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }d>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ 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 $c$: real or complex parameter

Keywords:

Mellin transform

Proof sketch:

Follows from (15.6.9) with $\lambda =a+b$.

Referenced by:

(15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)

Permalink:

http://dlmf.nist.gov/15.6.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►Note that (15.6.8) can be rewritten as a fractional integral. …

24: Bibliography O

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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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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

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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

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Tables of Mellin Transforms. Springer-Verlag, Berlin-New York.

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External Links:

ISBN 0-387-06942-9, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(iii), §13.23(i), §14.17(vi), §15.14, §18.17(ix), §20.10(i), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii).

Encodings:

BibTeX

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Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

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External Links:

FullText, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

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25: 1.15 Summability Methods

… ►where $F(t)$ is the Fourier transform of $f(x)$ (§1.14(i)). … ►

§1.15(vi) Fractional Integrals

►For $\mathrm{\Re }\alpha >0$ and $x\ge 0$, the Riemann-Liouville fractional integral of order $\alpha$ is defined by … ►

§1.15(vii) Fractional Derivatives

… ►Note that ${𝐷}^{1/2}𝐷\ne {𝐷}^{3/2}$. …

26: 3.11 Approximation Techniques

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Laplace Transform Inversion

►Numerical inversion of the Laplace transform (§1.14(iii)) … ►

Example. The Discrete Fourier Transform

… ►is called a discrete Fourier transform pair. ►

The Fast Fourier Transform

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27: 10.74 Methods of Computation

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§10.74(v) Continued Fractions

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Hankel Transform

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Spherical Bessel Transform

►The spherical Bessel transform is the Hankel transform (10.22.76) in the case when $\nu$ is half an odd positive integer. … ►

Kontorovich–Lebedev Transform

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28: 20.11 Generalizations and Analogs

… ►If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). …

29: 13.4 Integral Representations

… ►The fractional powers are continuous and assume their principal values at $t=\alpha$. …At this point the fractional powers are determined by $\mathrm{ph}t=\pi$ and $\mathrm{ph}\left(1+t\right)=0$. … ►

13.4.16 $$𝐌(a,b,-z)=\frac{1}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(b+t\right)}{z}^{t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

A&S Ref:

13.2.9 (in different form)

Permalink:

http://dlmf.nist.gov/13.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

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13.4.17 $$U(a,b,z)=\frac{z^{-a}}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(1+a-b+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(1+a-b\right)}{z}^{-t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

A&S Ref:

13.2.10

Referenced by:

§13.6(vi)

Permalink:

http://dlmf.nist.gov/13.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

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13.4.18 $$U(a,b,z)=\frac{z^{1-b}{\mathrm{e}}^z}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(b-1+t\right)\mathrm{\Gamma }\left(t\right)}{\mathrm{\Gamma }\left(a+t\right)}{z}^{-t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.4.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

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30: Bibliography R

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A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

BibTeX

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Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii).

Encodings:

BibTeX

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