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1: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

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§12.10(vi) Modifications of Expansions in Elementary Functions

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Modified Expansions

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2: Bibliography G

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Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

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ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§12.18.

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Computing the real parabolic cylinder functions $U(a,x)$, $V(a,x)$ . ACM Trans. Math. Software 32 (1), pp. 70–101.

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ISSN 0098-3500, Document, MathReview (Syamal K. Sen), ZentralBlatt

Referenced by:

§12.18.

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Algorithm 850: Real parabolic cylinder functions $U(a,x)$, $V(a,x)$ . ACM Trans. Math. Software 32 (1), pp. 102–112.

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ISSN 0098-3500, Document, MathReview (Syamal K. Sen), ZentralBlatt

Referenced by:

§12.18, 1st item.

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Algorithm 914: parabolic cylinder function $W(a,x)$ and its derivative. ACM Trans. Math. Software 38 (1), pp. Art. 6, 5.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§12.18, §12.18, 2nd item, §12.21(ii).

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Fast and accurate computation of the Weber parabolic cylinder function $W(a,x)$ . IMA J. Numer. Anal. 31 (3), pp. 1194–1216.

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ISSN 0272-4979, Document, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§12.18, §12.18.

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3: 12.19 Tables

§12.19 Tables

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Abramowitz and Stegun (1964, Chapter 19) includes $U(a,x)$ and $V(a,x)$ for $\pm a=0(.1)1(.5)5$, $x=0(.1)5$, 5S; $W(a,\pm x)$ for $\pm a=0(.1)1(1)5$, $x=0(.1)5$, 4-5D or 4-5S.

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Miller (1955) includes $W(a,x)$, $W(a,-x)$, and reduced derivatives for $a=-10(1)10$, $x=0(.1)10$, 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

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Fox (1960) includes modulus and phase functions for $W(a,x)$ and $W(a,-x)$, and several auxiliary functions for $x^{-1}=0(.005)0.1$, $a=-10(1)10$, 8S.

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Murzewski and Sowa (1972) includes $D_{-n}\left(x\right)$ $\left(=U(n-\frac{1}{2},x)\right)$ for $n=1(1)20$, $x=0(.05)3$, 7S.

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4: Bibliography S

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Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 18D.

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ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

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3rd item.

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Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

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Notes:

Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)

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ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.30(i), 3rd item.

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:

ZentralBlatt

Referenced by:

§12.13(i).

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

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On parabolic cylinder functions. J. Inst. Math. Appl. 4 (1), pp. 106–112.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§12.16.

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5: Software Index

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	Open Source													With Book						Commercial
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12 Parabolic Cylinder Functions
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

6: Bibliography F

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Generalized parabolic cylinder functions. Asymptotic Anal. 5 (6), pp. 517–531.

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External Links:

ISSN 0921-7134, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§12.15.

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

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Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

3rd item, 2nd item.

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

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7: Bibliography C

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The hypergeometric function and the $R$-function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.

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Notes:

International conference on special functions: theory and computation (Turin, 1984)

External Links:

ISSN 0373-1243, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.27(vi).

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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

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External Links:

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

Referenced by:

§12.16.

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:

Document

Referenced by:

5th item.

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8: 12.11 Zeros

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§12.11(i) Distribution of Real Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

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§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). … ►For the first zero of $U(a,x)$ we also have …

9: Bibliography O

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Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.

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External Links:

MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§12.10(i), §12.10(ii), §12.10(iii), §12.10(iv), §12.10(v), §12.10(vii), §12.10(vii), §12.11(iii), §12.11(iii), §12.14(ix), §12.14(ix), §12.14(ix), §12.14(xi), Ch.12, §18.16(v).

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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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External Links:

ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

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Formulae and Tables, The Modified Quotients of Cylinder Functions. Technical report Technical Report UDC 517.564.3:518.25, Vol. 4, Report of the Institute of Industrial Science, University of Tokyo, Institute of Industrial Science, Chiba City, Japan.

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Referenced by:

§10.29(i), §10.6(i).

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Modified quotients of cylinder functions. Math. Tables Aids Comput. 10, pp. 27–28.

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External Links:

ISSN 0891-6837, MathReview Entry, ZentralBlatt

Referenced by:

§10.29(i), §10.6(i).

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Scattering by a parabolic cylinder—a uniform asymptotic expansion. J. Math. Phys. 26 (4), pp. 854–860.

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External Links:

ISSN 0022-2488, Document, MathReview (Jean-Louis Guiraud)

Referenced by:

§12.17.

Encodings:

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10: 28.8 Asymptotic Expansions for Large $q$

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§28.8(ii) Sips’ Expansions

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§28.8(iii) Goldstein’s Expansions

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Barrett’s Expansions

… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►