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21—30 of 32 matching pages

22: Bibliography

… ►

M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

ⓘ

External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

… ►

A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

L. V. Ahlfors (1966) Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable. 2nd edition, McGraw-Hill Book Co., New York.

ⓘ

Notes:: A third edition was published in 1978 by McGraw-Hill, New York.
External Links:: MathReview (P. Lelong), ZentralBlatt
Referenced by:: §1.9(iii), §23.18.
Encodings:: BibTeX

… ►

S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

ⓘ

Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

…

23: 11.6 Asymptotic Expansions

… ►

11.6.1

\mathbf{K}_{\nu}\left(z\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{\Gamma% \left(k+\tfrac{1}{2}\right)(\tfrac{1}{2}z)^{\nu-2k-1}}{\Gamma\left(\nu+\tfrac{% 1}{2}-k\right)},

|\operatorname{ph}z|\leq\pi-\delta

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order, $k$ : nonnegative integer and $\delta$ : arbitrary small positive constant
A&S Ref:: 12.1.29
Referenced by:: §11.6(i), §11.6(i), §11.6(i), §11.6(i)
Permalink:: http://dlmf.nist.gov/11.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(i), §11.6 and Ch.11

… ►

11.6.2

\mathbf{M}_{\nu}\left(z\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}(-1)^{k+1}% \frac{\Gamma\left(k+\tfrac{1}{2}\right)(\tfrac{1}{2}z)^{\nu-2k-1}}{\Gamma\left% (\nu+\tfrac{1}{2}-k\right)},

|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order, $k$ : nonnegative integer and $\delta$ : arbitrary small positive constant
A&S Ref:: 12.2.6
Referenced by:: §11.6(i), §11.6(i), §11.6(i)
Permalink:: http://dlmf.nist.gov/11.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(i), §11.6 and Ch.11

… ►

11.6.3

\int_{0}^{z}\mathbf{K}_{0}\left(t\right)\,\mathrm{d}t-\frac{2}{\pi}(\ln\left(2% z\right)+\gamma)\sim\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(2k)!(2k-1% )!}{(k!)^{2}(2z)^{2k}},

|\operatorname{ph}z|\leq\pi-\delta

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable, $k$ : nonnegative integer and $\delta$ : arbitrary small positive constant
A&S Ref:: 12.1.32
Referenced by:: §11.6(i)
Permalink:: http://dlmf.nist.gov/11.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(i), §11.6 and Ch.11

… ►

11.6.5

\mathbf{H}_{\nu}\left(z\right),\mathbf{L}_{\nu}\left(z\right)\sim\frac{z}{\pi% \nu\sqrt{2}}\left(\frac{ez}{2\nu}\right)^{\nu},

|\operatorname{ph}\nu|\leq\pi-\delta

ⓘ

Symbols:: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order and $\delta$ : arbitrary small positive constant
Referenced by:: (11.11.7)
Permalink:: http://dlmf.nist.gov/11.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(ii), §11.6 and Ch.11

… ►

c_{3}(\lambda)=20\lambda^{-6}-4\lambda^{-4},

…

24: 9.9 Zeros

… ►However,

\operatorname{Bi}\left(z\right)

and

\operatorname{Bi}'\left(z\right)

each have an infinite number of complex zeros. They lie in the sectors

\tfrac{1}{3}\pi<\operatorname{ph}z<\tfrac{1}{2}\pi

and

-\tfrac{1}{2}\pi<\operatorname{ph}z<-\tfrac{1}{3}\pi

, and are denoted by

\beta_{k}

\beta^{\prime}_{k}

, respectively, in the former sector, and by

\overline{\beta_{k}}

\overline{\beta^{\prime}_{k}}

, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for

k=1,2,\ldots.

See §9.3(ii) for visualizations. ►For the distribution in

\mathbb{C}

of the zeros of

\operatorname{Ai}'\left(z\right)-\sigma\operatorname{Ai}\left(z\right)

, where

\sigma

is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014). … ►If

k

is regarded as a continuous variable, then … ►Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of

\operatorname{Bi}

and

\operatorname{Bi}'

in the upper half plane. …

25: 19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). … ►Complex values of the variables are allowed, with some restrictions in the case of

R_{J}

that are sufficient but not always necessary. … ►The incomplete integrals

R_{F}\left(x,y,z\right)

and

R_{G}\left(x,y,z\right)

can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to

R_{C}

, accompanied by two quadratically convergent series in the case of

R_{G}

; compare Carlson (1965, §§5,6). … ►The function

\operatorname{el2}\left(x,k_{c},a,b\right)

is computed by descending Landen transformations if

x

is real, or by descending Gauss transformations if

x

is complex (Bulirsch (1965b)). … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

26: Bibliography D

… ►

C. de la Vallée Poussin (1896a) Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction

\zeta(s)

de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

ⓘ

Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

… ►

B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

T. M. Dunster (1990b) Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §14.26, §2.8(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (1994b) Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (H. C. Howard), ZentralBlatt
Referenced by:: §2.8(vi).
Encodings:: BibTeX

… ►

A. Dzieciol, S. Yngve, and P. O. Fröman (1999) Coulomb wave functions with complex values of the variable and the parameters. J. Math. Phys. 40 (12), pp. 6145–6166.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Raj Wilson), ZentralBlatt
Referenced by:: §33.13.
Encodings:: BibTeX

27: 30.9 Asymptotic Approximations and Expansions

… ►

2^{20}\beta_{5}=-527q^{7}-61529q^{5}-10\;43961q^{3}-22\;41599q+32m^{2}(5739q^{% 5}+1\;27550q^{3}+2\;98951q)-2048m^{4}(355q^{3}+1505q)+65536m^{6}q.

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … ►The asymptotic behavior of

\lambda^{m}_{n}\left(\gamma^{2}\right)

and

a^{m}_{n,k}(\gamma^{2})

n\to\infty

in descending powers of

2n+1

is derived in Meixner (1944). …The behavior of

\lambda^{m}_{n}\left(\gamma^{2}\right)

for complex

\gamma^{2}

and large

|\lambda^{m}_{n}\left(\gamma^{2}\right)|

is investigated in Hunter and Guerrieri (1982). …

28: Bibliography K

… ►

A. A. Kapaev and A. V. Kitaev (1993) Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.

ⓘ

External Links:: ISSN 0377-9017, Document, MathReview (B. L. J. Braaksma), ZentralBlatt
Referenced by:: §32.11(i), §32.12(i).
Encodings:: BibTeX

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

ⓘ

External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

M. Kodama (2011) Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument. ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: 2nd item, §10.77(viii).
Encodings:: BibTeX

… ►

T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

ⓘ

External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

…

29: Bibliography I

… ►

Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of

J_{0}(z)-iJ_{1}(z)

and of Bessel functions

J_{m}(z)

of any real order

m

. Linear Algebra Appl. 194, pp. 35–70.

ⓘ

External Links:: ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

… ►

K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

… ►

M. E. H. Ismail (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt
Referenced by:: §1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

►

M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

Notes:: With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.
External Links:: ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt
Referenced by:: §18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

… ►

A. R. Its, A. S. Fokas, and A. A. Kapaev (1994) On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

ⓘ

External Links:: ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

…

30: 18.39 Applications in the Physical Sciences

… ►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form … ►

§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom

… ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►A major difficulty in such calculations, loss of precision, is addressed in Gautschi (2009) where use of variable precision arithmetic is discussed and employed. … ►where

s

is a real, positive, scaling factor, and

\it{l}

a non-negative integer. …