integral equations and representations

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11: Bibliography B

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Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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External Links:

ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§11.7(v), §11.7(v).

Encodings:

BibTeX

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Integral equations and exact solutions for the fourth Painlevé equation. Proc. Roy. Soc. London Ser. A 437, pp. 1–24.

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External Links:

ISSN 0962-8444, FullText, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(v), §32.2(iii).

Encodings:

BibTeX

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An Introduction to Linear Difference Equations. Dover Publications Inc., New York.

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Notes:

Unaltered republication of the original 1927 Harvard University edition.

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

Encodings:

BibTeX

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A new asymptotic representation for $\zeta (\frac{1}{2}+it)$ and quantum spectral determinants. Proc. Roy. Soc. London Ser. A 437, pp. 151–173.

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External Links:

ISSN 0962-8444, MathReview (Jack Quine), ZentralBlatt

Referenced by:

§25.9.

Encodings:

BibTeX

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Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

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External Links:

ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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12: 36.15 Methods of Computation

… ►(For the umbilics, representations as one-dimensional integrals (§36.2) are used.) … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. … ►

§36.15(v) Differential Equations

►For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).

13: 15.9 Relations to Other Functions

… ►This is a generalization of Jacobi polynomials (§18.3) and has the representation … ►

§15.9(v) Complete Elliptic Integrals

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15.9.24 $$K\left(k\right)=\frac{\pi }{2}F(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E24

Encodings:

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Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

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15.9.25 $$E\left(k\right)=\frac{\pi }{2}F(\genfrac{}{}{0pt}{}{-\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E25

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

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15.9.26 $$D\left(k\right)=\frac{\pi }{4}F(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{3}{2}}{2};k^2).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $D\left(k\right)$: complete elliptic integral of Legendre’s type and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E26

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

14: 9.13 Generalized Airy Functions

… ►Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: … ►Each of the functions $A_k(z,p)$ and $B_k(z,p)$ satisfies the differential equation …and the difference equation …

15: Errata

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The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint has been added. In (15.6.6), the constraint has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when , except (15.6.6) which holds for .”, has been removed.

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16: 24.7 Integral Representations

§24.7 Integral Representations

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§24.7(i) Bernoulli and Euler Numbers

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§24.7(ii) Bernoulli and Euler Polynomials

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Mellin–Barnes Integral

… ►For further integral representations see Prudnikov et al. (1986a, §§2.3–2.6) and Gradshteyn and Ryzhik (2000, Chapters 3 and 4).

17: 14.32 Methods of Computation

… ►In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. … ►

Numerical integration (§3.7) of the defining differential equations (14.2.2), (14.20.1), and (14.21.1).

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Quadrature (§3.5) of the integral representations given in §§14.12, 14.19(iii), 14.20(iv), and 14.25; see Segura and Gil (1999) and Gil et al. (2000).

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18: 25.14 Lerch’s Transcendent

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25.14.5 $$\mathrm{\Phi }(z,s,a)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1-z{\mathrm{e}}^{-x}}dx,$$ $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $z=1$; $\mathrm{\Re }s>0$, $\mathrm{\Re }a>0$ if $z\in ℂ\setminus [1,\mathrm{\infty })$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $[a,b)$: half-closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $\setminus$: set subtraction, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integralrepresentation

Source:

Erdélyi et al. (1953a, (1.11.3), p. 27)

Referenced by:

Erratum (V1.1.4) for Equation (25.14.5)

Permalink:

http://dlmf.nist.gov/25.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

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25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integralrepresentation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

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Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

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19: 22.16 Related Functions

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Integral Representation

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Integral Representations

►For , …See Figure 22.16.2. … ►In Equations (22.16.21)–(22.16.23), …

20: 15.6 Integral Representations

§15.6 Integral Representations

►The function $𝐅(a,b;c;z)$ (not $F(a,b;c;z)$) has the following integral representations: … ►

15.6.9 $$𝐅(a,b;c;z)={\int }_0^1\frac{t^{d-1}{(1-t)}^{c-d-1}}{{(1-zt)}^{a+b-\lambda }}𝐅(\genfrac{}{}{0pt}{}{\lambda -a,\lambda -b}{d};zt)𝐅(\genfrac{}{}{0pt}{}{a+b-\lambda ,\lambda -d}{c-d};\frac{(1-t)z}{1-zt})dt,$$ ; $\lambda \in ℂ$, $\mathrm{\Re }c>\mathrm{\Re }d>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: 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

Notes:

With $\lambda =a+b$, this equation specializes to (15.6.8).

Referenced by:

(15.6.8), §15.6, §15.6, Erratum (V1.0.18) for Equation (15.6.9), 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.E9

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.

Addition (effective with 1.0.18):

The constraint $\lambda \in ℂ$ was added.

See also:

Annotations for §15.6 and Ch.15

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

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Figure 15.6.1: $t$-plane. … Magnify