integral representation of solutions

(0.004 seconds)

11—20 of 42 matching pages

11: 32.10 Special Function Solutions

§32.10 Special Function Solutions

… ►with solution … ►For determinantal representations see Forrester and Witte (2002) and Okamoto (1987c). … ►which has the solution … ►

12: Bibliography M

… ►

O. I. Marichev (1984) On the Representation of Meijer’s

G

-Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.

ⓘ

External Links:: MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §16.21.
Encodings:: BibTeX

… ►

P. Maroni (1995) An integral representation for the Bessel form. J. Comput. Appl. Math. 57 (1-2), pp. 251–260.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt
Referenced by:: §18.34(ii).
Encodings:: BibTeX

… ►

I. Mező (2020) An integral representation for the Lambert

W

function.

ⓘ

External Links:: 2012.02480
Referenced by:: (4.13.16), §4.13.
Encodings:: BibTeX

… ►

G. F. Miller (1966) On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.

ⓘ

External Links:: ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

►

J. C. P. Miller (1946) The Airy Integral, Giving Tables of Solutions of the Differential Equation

y^{\prime\prime}=xy

. British Association for the Advancement of Science, Mathematical Tables Part-Vol. B, Cambridge University Press, Cambridge.

ⓘ

External Links:: MathReview (C. J. Bouwkamp), ZentralBlatt
Referenced by:: 1st item, 1st item, §9.2(i), §9.2(ii), §9.2(v), §9.4, (9.6.2), (9.6.3), (9.6.4), (9.6.5), (9.6.6), (9.6.7), (9.6.8), (9.6.9), §9.6(i), §9.8(i), §9.8(ii), §9.8(iv), §9.9(ii), Ch.9.
Encodings:: BibTeX

…

13: Bibliography L

… ►

R. E. Langer (1934) The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

Y. T. Li and R. Wong (2008) Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.

ⓘ

External Links:: ISSN 1534-0392, MathReview Entry, ZentralBlatt
Referenced by:: §1.17(iv), (1.18.57).
Encodings:: BibTeX

… ►

J. C. Light and T. Carrington Jr. (2000) Discrete-variable representations and their utilization. In Advances in Chemical Physics, pp. 263–310.

ⓘ

External Links:: ISBN 9780470141731, Document, Link
Referenced by:: §18.38(i).
Encodings:: BibTeX

… ►

J. L. López and N. M. Temme (1999b) Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (A. G. Law), ZentralBlatt
Referenced by:: §18.34(iii), §24.11, §24.16(i), §24.16(i).
Encodings:: BibTeX

… ►

T. A. Lowdon (1970) Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

ⓘ

External Links:: Document, MathReview (T. Erber), ZentralBlatt
Referenced by:: §12.12, §12.16.
Encodings:: BibTeX

…

14: 13.27 Mathematical Applications

… ►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. …Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. … ►

15: Bibliography G

… ►

L. Gårding (1947) The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

ⓘ

External Links:: FullText, MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §35.2, §35.3(ii).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2004c) Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §12.18.
Encodings:: BibTeX

… ►

H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

ⓘ

External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

K. I. Gross and R. A. Kunze (1976) Bessel functions and representation theory. I. J. Functional Analysis 22 (2), pp. 73–105.

ⓘ

External Links:: Document, MathReview (E. A. Thieleker), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

… ►

E. Grosswald (1985) Representations of Integers as Sums of Squares. Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

…

16: Bibliography C

… ►

J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.

ⓘ

External Links:: MathReview (T. Hayashida)
Referenced by:: §27.13(ii).
Encodings:: BibTeX

… ►

C. Chiccoli, S. Lorenzutta, and G. Maino (1990b) On a Tricomi series representation for the generalized exponential integral. Internat. J. Comput. Math. 31, pp. 257–262.

ⓘ

Notes:: Includes Fortran code
External Links:: Document, ZentralBlatt
Referenced by:: §8.28(vi).
Encodings:: BibTeX

… ►

M. W. Coffey (2008) On some series representations of the Hurwitz zeta function. J. Comput. Appl. Math. 216 (1), pp. 297–305.

ⓘ

External Links:: ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §25.11(iv), §25.11(iv).
Encodings:: BibTeX

… ►

H. S. Cohl and R. S. Costas-Santos (2020) Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).

ⓘ

External Links:: Link, ISSN 2073-8994, Document, ZentralBlatt
Referenced by:: §14.6(ii), §14.6(ii), Erratum (V1.1.1) for Subsection 14.6(ii).
Encodings:: BibTeX

… ►

H. S. Cohl (2011) On parameter differentiation for integral representations of associated Legendre functions. SIGMA Symmetry Integrability Geom. Methods Appl. 7, pp. Paper 050, 16.

ⓘ

External Links:: ISSN 1815-0659, Document, FullText, MathReview (Michael Doschoris), ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

…

17: 11.9 Lommel Functions

… ►Provided that

\mu\pm\nu\neq-1,-3,-5,\dots

, (11.9.1) has the general solution … ►Another solution of (11.9.1) that is defined for all values of

\mu

and

\nu

S_{{\mu},{\nu}}\left(z\right)

, where … ►For uniform asymptotic expansions, for large

\nu

and fixed

\mu=-1,0,1,2,\dots

, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … ►For collections of integral representations and integrals see Apelblat (1983, §12.17), Babister (1967, p. 85), Erdélyi et al. (1954a, §§4.19 and 5.17), Gradshteyn and Ryzhik (2000, §6.86), Marichev (1983, p. 193), Oberhettinger (1972, pp. 127–128, 168–169, and 188–189), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105–106 and 191–192), Oberhettinger and Badii (1973, §2.14), Prudnikov et al. (1990, §§1.6 and 2.9), Prudnikov et al. (1992a, §3.34), and Prudnikov et al. (1992b, §3.32).

18: 9.11 Products

… ►where

w_{1}

and

w_{2}

are any solutions of (9.2.1). … ►

§9.11(iii) Integral Representations

… ►For an integral representation of the Dirac delta involving a product of two

\operatorname{Ai}

functions see §1.17(ii). ►For further integral representations see Reid (1995, 1997a, 1997b). … ►Let

w_{1},w_{2}

be any solutions of (9.2.1), not necessarily distinct. …

19: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

… ►

36.11.1

t_{1}(\mathbf{x})<t_{2}(\mathbf{x})<\cdots<t_{j_{\max}}(\mathbf{x}),

ⓘ

Symbols:: $t_{j}(\mathbf{x})$ : solutions
Permalink:: http://dlmf.nist.gov/36.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11 and Ch.36

►and far from the bifurcation set, the cuspoid canonical integrals are approximated by ►

36.11.2

\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right|^{-1/2}(1+o\left(1% \right)).

ⓘ

Symbols:: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $t$ : variable, $K$ : codimension and $t_{j}(\mathbf{x})$ : solutions
Permalink:: http://dlmf.nist.gov/36.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11 and Ch.36

… ►

36.11.7

\Psi^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(% \frac{4}{27}iz^{3}\right)+o\left(1\right)\right),

z\to\pm\infty

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $z$ : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

…

20: 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

\Phi

, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. … ►For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).