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1: 13.4 Integral Representations

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13.4.2 $$\mathbf{M}(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(b-c\right)}{\int }_0^1\mathbf{M}(a,c,zt)t^{c-1}{(1-t)}^{b-c-1}dt,$$ $\mathrm{\Re }b>\mathrm{\Re }c>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $c$: arbitrary

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E2

Encodings:

pMML, png, TeX

See also:

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

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13.4.6 $$U(a,b,z)=\frac{{(-1)}^n{z}^{1-b-n}}{\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{\mathbf{M}(b-a,b,t){\mathrm{e}}^{-t}{t}^{b+n-1}}{t+z}dt,$$ , $n=0,1,2,\mathrm{\dots }$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E6

Encodings:

pMML, png, TeX

See also:

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

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2: 10.43 Integrals

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§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\mathrm{\infty })$

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3: 13.16 Integral Representations

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13.16.2 $$M_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(1+2\mu \right)z^{\lambda }}{\mathrm{\Gamma }\left(1+2\mu -2\lambda \right)\mathrm{\Gamma }\left(2\lambda \right)}{\int }_0^1{M}_{\kappa -\lambda ,\mu -\lambda }\left(zt\right){\mathrm{e}}^{\frac{1}{2}z(t-1)}{t}^{\mu -\lambda -\frac{1}{2}}{(1-t)}^{2\lambda -1}dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\mathrm{\Re }\lambda >0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.23(v)

Permalink:

http://dlmf.nist.gov/13.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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4: 9.12 Scorer Functions

… ►If $\zeta =\frac{2}{3}{z}^{3/2}$ or $\frac{2}{3}{x}^{3/2}$, and $K_{1/3}$ is the modified Bessel function (§10.25(ii)), then …

5: 36.9 Integral Identities

… ►For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980). …

6: 10.22 Integrals

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§10.22(iii) Integrals over the Interval $(x,\mathrm{\infty })$

… ►Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

7: 2.8 Differential Equations with a Parameter

… ►and for simplicity $\xi$ is assumed to range over a finite or infinite interval $({\alpha }_1,{\alpha }_2)$ with , ${\alpha }_2>0$. …

8: 2.3 Integrals of a Real Variable

… ►If, in addition, $q(t)$ is infinitely differentiable on $[0,\mathrm{\infty })$ and … ►assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. …Alternatively, assume $b=\mathrm{\infty }$, $q(t)$ is infinitely differentiable on $[a,\mathrm{\infty })$, and each of the integrals $\int {\mathrm{e}}^{\mathrm{i}xt}{q}^{(s)}(t)dt$, $s=0,1,2,\mathrm{\dots }$, converges as $t\to \mathrm{\infty }$ uniformly for all sufficiently large $x$. … ►Assume that $q(t)$ again has the expansion (2.3.7) and this expansion is infinitely differentiable, $q(t)$ is infinitely differentiable on $(0,\mathrm{\infty })$, and each of the integrals $\int {\mathrm{e}}^{\mathrm{i}xt}{q}^{(s)}(t)dt$, $s=0,1,2,\mathrm{\dots }$, converges at $t=\mathrm{\infty }$, uniformly for all sufficiently large $x$. … ►

On $(a,b)$, $p(t)$ and $q(t)$ are infinitely differentiable and $p^{\prime }(t)>0$.

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9: 1.4 Calculus of One Variable

… ►If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^n(I)$. …When $n$ is unbounded, $f$ is infinitely differentiable on $I$ and we write $f\in C^{\mathrm{\infty }}(I)$. … ►

Infinite Integrals

… ►With , the total variation of $f(x)$ on a finite or infinite interval $(a,b)$ is …where the supremum is over all sets of points in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. …

10: Bibliography I

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IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-0-7381-9720-3, Document

Referenced by:

§3.1(ii), §3.1(ii), 4th Erratum.

Encodings:

BibTeX

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IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-1-5044-4602-0, Document

Referenced by:

§3.1(ii), §3.1(ii), 4th Erratum.

Encodings:

BibTeX

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

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External Links:

ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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