integral equations and representations

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31: 7.7 Integral Representations

§7.7 Integral Representations

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§7.7(i) Error Functions and Dawson’s Integral

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§7.7(ii) Auxiliary Functions

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Mellin–Barnes Integrals

… ►For other integral representations see Erdélyi et al. (1954a, vol. 1, pp. 265–267, 270), Ng and Geller (1969), Oberhettinger (1974, pp. 246–248), and Oberhettinger and Badii (1973, pp. 371–377).

32: 9.12 Scorer Functions

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§9.12(i) Differential Equation

… ►where … ►

§9.12(vii) Integral Representations

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Integrals

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33: 9.17 Methods of Computation

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§9.17(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods. … ►

§9.17(iii) Integral Representations

►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. … ►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …

34: 15.19 Methods of Computation

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§15.19(ii) Differential Equation

►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. …However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. ►

§15.19(iii) Integral Representations

►The representation (15.6.1) can be used to compute the hypergeometric function in the sector . …

35: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

… ►The function $s(ϵ,\mathrm{\ell };r)$ has the following properties: ►

33.14.13 $${\int }_0^{\mathrm{\infty }}s({ϵ}_1,\mathrm{\ell };r)s({ϵ}_2,\mathrm{\ell };r)dr=\delta \left({ϵ}_1-{ϵ}_2\right),$$ ${ϵ}_1,{ϵ}_2>0$,

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Symbols:

$s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.0.24):

The constraint ${ϵ}_1,{ϵ}_2>0$ was added.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

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33.14.14 $${\varphi }_{n,\mathrm{\ell }}(r)={(-1)}^{\mathrm{\ell }+1+n}{(2/n^3)}^{1/2}s(-1/n^2,\mathrm{\ell };r)=\frac{{(-1)}^{\mathrm{\ell }+1+n}}{n^{\mathrm{\ell }+2}}{\left(\frac{(n-\mathrm{\ell }-1)!}{(n+\mathrm{\ell })!}\right)}^{1/2}{(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/n}{L}_{n-\mathrm{\ell }-1}^{(2\mathrm{\ell }+1)}\left(2r/n\right)$$

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Defines:

${\varphi }_{n,\mathrm{\ell }}(r)$: function (locally)

Symbols:

$s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer and $r$: real variable

Referenced by:

(33.14.14), (33.14.15), §33.14(iv), §33.22(v), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E14

Encodings:

pMML, png, TeX

Addition (effective with 1.0.24):

For the second equation in (33.14.14) first, via (33.14.9) and (33.14.4), express ${\varphi }_{n,\mathrm{\ell }}(r)$ in terms of Whittaker functions and then use (13.18.17).

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

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33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and ${\varphi }_{n,\mathrm{\ell }}(r)$: function

Proof sketch:

Derivable from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E15

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.24):

This equation, originally written as a definite integral ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was rewritten more generally as an orthonormality relation.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

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36: 9.5 Integral Representations

§9.5 Integral Representations

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§9.5(i) Real Variable

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9.5.1 $$\mathrm{Ai}\left(x\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\cos \left(\frac{1}{3}{t}^3+xt\right)dt.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Source:

Olver (1997b, p. 53)

A&S Ref:

10.4.32 (in different form)

Permalink:

http://dlmf.nist.gov/9.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(i), §9.5 and Ch.9

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§9.5(ii) Complex Variable

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9.5.6 $$\mathrm{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{t^3}{3}-\frac{z^3}{3t^3}\right)dt,$$ .

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Source:

Reid (1995, (5.4), p. 170, with change of variable and using analytic continuation to complex $z$)

Referenced by:

Erratum (V1.0.15) for Equation (9.5.6)

Permalink:

http://dlmf.nist.gov/9.5.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

The validity constraint was added.

See also:

Annotations for §9.5(ii), §9.5 and Ch.9

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37: 10.74 Methods of Computation

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§10.74(ii) Differential Equations

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§10.74(iii) Integral Representations

►For evaluation of the Hankel functions $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ for complex values of $\nu$ and $z$ based on the integral representations (10.9.18) see Remenets (1973). … ►The integral representation used is based on (10.32.8). … ►

§10.74(vii) Integrals

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38: 22.15 Inverse Functions

… ►With real variables, the solutions of the equations … ►Equations (22.15.1) and (22.15.4), for $\mathrm{arcsn}(x,k)$, are equivalent to (22.15.12) and also to … ►

§22.15(ii) Representations as Elliptic Integrals

… ►Other integrals, for example, … ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …

39: 10.32 Integral Representations

§10.32 Integral Representations

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§10.32(i) Integrals along the Real Line

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§10.32(ii) Contour Integrals

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§10.32(iii) Products

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§10.32(iv) Compendia

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40: 13.16 Integral Representations

§13.16 Integral Representations

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§13.16(i) Integrals Along the Real Line

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§13.16(ii) Contour Integrals

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§13.16(iii) Mellin–Barnes Integrals

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