integral transforms in terms of

SearchAdvancedHelp

(0.009 seconds)

21—30 of 48 matching pages

21: Bibliography T

… ►

Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

ⓘ

External Links:

ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§13.8(iii), §13.8(iii).

Encodings:

BibTeX

… ►

Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

ⓘ

External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

BibTeX

… ►

A note on the real zeros of the incomplete gamma function. Integral Transforms Spec. Funct. 23 (6), pp. 445–453.

ⓘ

External Links:

ISSN 1065-2469, Document, FullText, MathReview (Feng Qi), ZentralBlatt

Referenced by:

§8.13(i), §8.13(i).

Encodings:

BibTeX

… ►

Theory of the Fresnel integral. USSR Comput. Math. and Math. Phys. 9 (4), pp. 271–279.

ⓘ

Notes:

Translation of the paper in Zh. Vychisl. Mat. Mat. Fiz., Vol. 9, No. 4, 938-944, 1969

External Links:

Document, ZentralBlatt

Referenced by:

§7.13(iv).

Encodings:

BibTeX

22: 2.8 Differential Equations with a Parameter

… ►An alternative way of representing the error terms in (2.8.15) and (2.8.16) is as follows. … ►

§2.8(iv) Case III: Simple Pole

… ►Again, an alternative way of representing the error terms in (2.8.29) and (2.8.30) is by means of envelope functions. … ►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. … ►For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv). …

23: 22.20 Methods of Computation

… ►A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument $z$ and the modulus $k$ is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. … ►

§22.20(iii) Landen Transformations

►By application of the transformations given in §§22.7(i) and 22.7(ii), $k$ or $k^{\prime }$ can always be made sufficently small to enable the approximations given in §22.10(ii) to be applied. … ►From the first two terms in (22.10.6) we find $\mathrm{dn}(0.19,\frac{1}{19})=0.999951$. … ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. …

24: Guide to Searching the DLMF

… ►

Terms, Phrases and Expressions

►Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators: ►

a textual word, a number, or a math symbol.

… ►If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase. … ►

Single-letter terms

…

25: Bibliography C

… ►

Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt

Referenced by:

§16.24(ii).

Encodings:

BibTeX

… ►

Landen Transformations of Integrals. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Appl. Math., Vol. 124, pp. 75–94.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.22(iii).

Encodings:

BibTeX

… ►

Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

ⓘ

External Links:

MathReview (I. I. Hirschman, Jr.), ZentralBlatt

Referenced by:

§12.16.

Encodings:

BibTeX

… ►

On a generalization of the generating function for Gegenbauer polynomials. Integral Transforms Spec. Funct. 24 (10), pp. 807–816.

ⓘ

External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.28(ii), §14.28(ii).

Encodings:

BibTeX

… ►

Evaluation of multidimensional canonical integrals in semiclassical collision theory. Molecular Phys. 26 (6), pp. 1371–1377.

ⓘ

External Links:

Document

Referenced by:

§36.8.

Encodings:

BibTeX

…

26: 15.19 Methods of Computation

… ►For $z\in ℝ$ it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval $[0,\frac{1}{2}]$. ►For $z\in ℂ$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. … ►

§15.19(iii) Integral Representations

… ►When $\mathrm{\Re }z>\frac{1}{2}$ it is better to begin with one of the linear transformations (15.8.4), (15.8.7), or (15.8.8). … …

27: Bibliography M

… ►

Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables. Ellis Horwood Ltd./John Wiley & Sons, Inc, Chichester/New York.

ⓘ

Notes:

Edited by F. D. Gakhov, Translated from the Russian by L. W. Longdon

External Links:

ISBN 0-85312-538-7, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.32(iv), §10.43(vi), §10.9(iv), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iii), §13.23(i), §14.17(vi), §15.14, §18.17(ix), §6.14(iii), §7.14(iii), §8.14.

Encodings:

BibTeX

… ►

Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. In Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), Vol. 49, pp. 169–178.

ⓘ

External Links:

ISSN 0377-0427, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

… ►

Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.

ⓘ

External Links:

ISSN 0020-2932, Document, MathReview (G. M. Habibullah), ZentralBlatt

Referenced by:

§2.6(ii).

Encodings:

BibTeX

… ►

A $q$-analog of a Whipple’s transformation for hypergeometric series in $U(n)$ . Adv. Math. 108 (1), pp. 1–76.

ⓘ

External Links:

ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

… ►

Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.

ⓘ

External Links:

Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§29.11, §29.7(ii).

Encodings:

BibTeX

…

28: 3.5 Quadrature

… ►In particular, when $k=\mathrm{\infty }$ the error term is an exponentially-small function of $1/h$, and in these circumstances the composite trapezoidal rule is exceptionally efficient. … ►Oscillatory integral transforms are treated in Wong (1982) by a method based on Gaussian quadrature. … ►In fact from (7.14.4) and the inversion formula for the Laplace transform (§1.14(iii)) we have … ► ►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)). …

29: 31.10 Integral Equations and Representations

… ►The solutions of (31.10.8) are given in terms of the Riemann $P$-symbol (see §15.11(i)) as … ►For suitable choices of the branches of the $P$-symbols in (31.10.9) and the contour $C$, we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution). … ►This equation can be solved in terms of cylinder functions ${𝒞}_{\nu }\left(z\right)$ (§10.2(ii)): … ►

Transformation of Independent Variable

… ►This equation can be solved in terms of hypergeometric functions (§15.11(i)): …

30: 7.7 Integral Representations

§7.7 Integral Representations

►

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

►Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions. … ►

§7.7(ii) Auxiliary Functions

… ►

Mellin–Barnes Integrals

…