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31: 10.49 Explicit Formulas

§10.49 Explicit Formulas

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§10.49(i) Unmodified Functions

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§10.49(ii) Modified Functions

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§10.49(iii) Rayleigh’s Formulas

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§10.49(iv) Sums or Differences of Squares

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32: 14.9 Connection Formulas

§14.9 Connection Formulas

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§14.9(i) Connections Between ${𝖯}_{\nu }^{\pm \mu }\left(x\right)$, ${𝖯}_{-\nu -1}^{\pm \mu }\left(x\right)$, ${𝖰}_{\nu }^{\pm \mu }\left(x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(x\right)$

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§14.9(ii) Connections Between ${𝖯}_{\nu }^{\pm \mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$

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§14.9(iii) Connections Between $P_{\nu }^{\pm \mu }\left(x\right)$, $P_{-\nu -1}^{\pm \mu }\left(x\right)$, ${𝑸}_{\nu }^{\pm \mu }\left(x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(x\right)$

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§14.9(iv) Whipple’s Formula

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33: 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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15.6.3 $$𝐅(a,b;c;z)={\mathrm{e}}^{-b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(1-b\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(c-b\right)}{\int }_{\mathrm{\infty }}^{(0+)}\frac{t^{b-1}{(t+1)}^{a-c}}{{(t-zt+1)}^a}dt,$$ ; $b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }\left(c-b\right)>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: 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, §15.6

Permalink:

http://dlmf.nist.gov/15.6.E3

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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15.6.4 $$𝐅(a,b;c;z)={\mathrm{e}}^{-b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(1-b\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(c-b\right)}{\int }_1^{(0+)}\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }\left(c-b\right)>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: 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, §15.6

Permalink:

http://dlmf.nist.gov/15.6.E4

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

ⓘ

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

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34: 3.4 Differentiation

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Two-Point Formula

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Three-Point Formula

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Four-Point Formula

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Five-Point Formula

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Six-Point Formula

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35: 36.5 Stokes Sets

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§36.5(ii) Cuspoids

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§36.5(iii) Umbilics

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36: 10.4 Connection Formulas

§10.4 Connection Formulas

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37: 6.18 Methods of Computation

… ►Also, other ranges of $\mathrm{ph}z$ can be covered by use of the continuation formulas of §6.4. … ►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968). …

38: About MathML

… ►MathML allows us to present the mathematics independent on your screen size and resolution, enabling adjustment to enlarge or shrink formula, as well as providing opportunities for making the material accessible for those with disabilities. …

39: Foreword

… ►In 1964 the National Institute of Standards and Technology published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. …

40: Preface

… ►The project had two equally important goals: to develop an authoritative replacement for the highly successful Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, published in 1964 by the National Bureau of Standards (M. … ►All of the mathematical information contained in the Handbook is also contained in the DLMF, along with additional features such as more graphics, expanded tables, and higher members of some families of formulas; in consequence, in the Handbook there are occasional gaps in the numbering sequences of equations, tables, and figures. … ►Among these capabilities are: a facility to allow users to download LaTeX and MathML encodings of every formula into document processors and software packages (eventually, a fully semantic downloading capability may be possible); a search engine that allows users to locate formulas based on queries expressed in mathematical notation; and user-manipulable 3-dimensional color graphics. …