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41: Bibliography L

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Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

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External Links:

ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(ii).

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

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

Encodings:

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Numerical Solution of Linear Difference Equations. NBSIR Technical Report 80-1976, National Bureau of Standards, Gaithersburg, MD 20899.

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Notes:

(Available from National Technical Information Service, Springfield, VA 22161.)

Referenced by:

§16.25, §3.6(vii).

Encodings:

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Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.

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External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.24(ii).

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Linear prediction of bandlimited processes with flat spectral densities. IEEE Trans. Signal Process. 49 (7), pp. 1564–1569.

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External Links:

ISSN 1053-587X, Document, MathReview, ZentralBlatt

Referenced by:

§30.15(v).

Encodings:

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42: 31.16 Mathematical Applications

§31.16 Mathematical Applications

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§31.16(i) Uniformization Problem for Heun’s Equation

… ► thesis “Inversion problem for a second-order linear differential equation with four singular points”. … ►

31.16.1 $${\mathrm{𝐻𝑝}}_{n,m}\left(x\right){\mathrm{𝐻𝑝}}_{n,m}\left(y\right)=\sum _{j=0}^n{A}_j{\sin }^{2j}\theta P_{n-j}^{(\gamma +\delta +2j-1,ϵ-1)}\left(\cos \left(2\theta \right)\right)P_j^{(\delta -1,\gamma -1)}\left(\cos \left(2\varphi \right)\right),$$

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Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $\theta$: angle, $\varphi$: angle, $x$: variable, $y$: variable and $A_j$: coefficients

Referenced by:

§31.16(ii), §31.16(ii)

Permalink:

http://dlmf.nist.gov/31.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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43: 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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Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.13(i), §7.13(ii), 1st item.

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Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

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External Links:

FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.10.

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Spheroidal Wave Functions. Stanford University Press, Stanford, CA.

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External Links:

MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§30.1, §30.10, 2nd item, Ch.30.

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Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63 (3), pp. 301–321.

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External Links:

ISSN 0025-5831, Document, ZentralBlatt

Referenced by:

§32.2(iv).

Encodings:

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44: 35.10 Methods of Computation

§35.10 Methods of Computation

… ►See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). … ►These algorithms are extremely efficient, converge rapidly even for large values of $m$, and have complexity linear in $m$.

45: Bibliography R

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On the partition function p(n). Proc. London Math. Soc. (2) 43 (4), pp. 241–254.

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External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(iii).

Encodings:

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

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Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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External Links:

MathReview (H. H. Martens), ZentralBlatt

Referenced by:

§20.11(iv), §20.12(ii).

Encodings:

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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

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Notes:

English translation, AEC-TR-4491, United States Atomic Energy Commission

External Links:

MathReview (A. S. Householder)

Referenced by:

§3.11(iii).

Encodings:

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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External Links:

Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

Encodings:

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46: Bibliography K

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Quasi-linear Stokes phenomenon for the Painlevé first equation. J. Phys. A 37 (46), pp. 11149–11167.

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External Links:

ISSN 0305-4470, Document, MathReview (W. Balser), ZentralBlatt

Referenced by:

§32.12(i).

Encodings:

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Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.

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External Links:

ISSN 0002-9890, FullText, MathReview (J. G. Herriot), ZentralBlatt

Referenced by:

§3.8(iii).

Encodings:

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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

Encodings:

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Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.

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Notes:

Single-precision Fortran, accuracy 12-14S

External Links:

Document, Code

Referenced by:

1st item.

Encodings:

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An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1), pp. 3–43.

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External Links:

ISSN 0747-7171, MathReview, ZentralBlatt

Referenced by:

§31.14(ii).

Encodings:

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47: Bibliography H

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Certain integrals that contain a probability function and degenerate hypergeometric functions. Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).

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External Links:

MathReview (Wazir Hasan Abdi)

Referenced by:

§7.14(iii).

Encodings:

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Some integrals that contain a probability function and hypergeometric functions. Bul. Akad. Štiince RSS Moldoven 1970 (1), pp. 49–62 (Russian).

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External Links:

MathReview (R. G. Langebartel)

Referenced by:

§7.14(iii).

Encodings:

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

Encodings:

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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 2nd edition, Prentice Hall Inc., Upper Saddle River, NJ.

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External Links:

ISBN 0-13-041408-5; 978-0-13-041408-3, MathReview (W. P. Ziemer), ZentralBlatt

Referenced by:

§1.6(i).

Encodings:

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48: Bibliography N

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Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

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External Links:

ISSN 0031-9015, Document, MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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Referenced by:

§2.8(vi).

Encodings:

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Inequalities and bounds for the incomplete gamma function. Results Math. 63 (3-4), pp. 1209–1214.

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External Links:

ISSN 1422-6383, Document, FullText, MathReview (Pierpaolo Natalini), ZentralBlatt

Referenced by:

§8.10, §8.10.

Encodings:

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Jacobian Elliptic Functions. 2nd edition, Clarendon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§20.1, Notations T, Notations T, Notations T, Notations T.

Encodings:

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Hypergeometric functions. Acta Math. 94, pp. 289–349.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.10, §16.8(ii), §16.8(ii).

Encodings:

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49: 21.7 Riemann Surfaces

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§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces

►In almost all applications, a Riemann theta function is associated with a compact Riemann surface. … ►Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$. …is a Riemann matrix and it is used to define the corresponding Riemann theta function. … ► …

50: 28.34 Methods of Computation

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Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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§28.34(iv) Modified Mathieu Functions

►For the modified functions we have: … ►

Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.