expansions of solutions in series of

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31: Errata

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§4.13 has been enlarged. The Lambert $W$-function is multi-valued and we use the notation $W_k\left(x\right)$, $k\in \mathbb{Z}$, for the branches. The original two solutions are identified via $\mathrm{Wp}\left(x\right)=W_0\left(x\right)$ and $\mathrm{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$.

Other changes are the introduction of the Wright $\omega$-function and tree $T$-function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{d^{n}W}{{dz}^n}$, additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$-functions in the end of the section.

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32: 29.6 Fourier Series

§29.6 Fourier Series

… ►When $\nu \ne 2n$, where $n$ is a nonnegative integer, it follows from §2.9(i) that for any value of $H$ the system (29.6.4)–(29.6.6) has a unique recessive solution $A_0,A_2,A_4,\mathrm{\dots }$; furthermore … ►In the special case $\nu =2n$, $m=0,1,\mathrm{\dots },n$, there is a unique nontrivial solution with the property $A_{2p}=0$, $p=n+1,n+2,\mathrm{\dots }$. This solution can be constructed from (29.6.4) by backward recursion, starting with $A_{2n+2}=0$ and an arbitrary nonzero value of $A_{2n}$, followed by normalization via (29.6.5) and (29.6.6). … ►An alternative version of the Fourier series expansion (29.6.1) is given by …

33: Bibliography V

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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.12, §19.5.

Encodings:

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Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:

Document

Referenced by:

§14.31(i).

Encodings:

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Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:

ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

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On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

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Notes:

In Russian. English translation: Differential Equations 1(1), pp. 58–59.

External Links:

MathReview, ZentralBlatt

Referenced by:

§32.8(ii).

Encodings:

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34: 28.15 Expansions for Small $q$

§28.15 Expansions for Small $q$

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§28.15(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$

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28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

Encodings:

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See also:

Annotations for §28.15(i), §28.15 and Ch.28

►Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a={\lambda }_{\nu }\left(q\right)$: … ►

§28.15(ii) Solutions ${\mathrm{me}}_{\nu }(z,q)$

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35: Bibliography H

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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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Divergent Series. Clarendon Press, Oxford.

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Notes:

Reprinted by the American Mathematical Society in 2000.

External Links:

MathReview (R. P. Agnew), ZentralBlatt

Referenced by:

§1.15(ii), §1.15(viii).

Encodings:

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Applied and Computational Complex Analysis. Vol. 1: Power Series—Integration—Conformal Mapping—Location of Zeros. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-471-37244-7, MathReview (M. Marden), ZentralBlatt

Referenced by:

§1.10(vii), §1.9(vi), Ch.3.

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Solutions of Poisson’s equation in channel-like geometries. Comput. Phys. Comm. 115 (1), pp. 45–68.

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External Links:

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

Referenced by:

§14.31(i).

Encodings:

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Asymptotic expansions of Mathieu functions in wave mechanics. J. Comput. Phys. 21 (3), pp. 319–325.

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External Links:

Document, MathReview (John P. Coleman)

Referenced by:

5th item.

Encodings:

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36: 15.19 Methods of Computation

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§15.19(i) Maclaurin Expansions

►The Gauss series (15.2.1) converges for . … ►However, by appropriate choice of the constant $z_0$ in (15.15.1) we can obtain an infinite series that converges on a disk containing $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. … ►As noted in §3.7(ii), the integration path should be chosen so that the wanted solution grows in magnitude at least as fast as all other solutions. … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …

37: Bibliography J

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

Encodings:

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

A second edition was published in 2010.

External Links:

ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt

Referenced by:

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).

Encodings:

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Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.

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Notes:

Third edition published in 1965 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI

External Links:

MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§26.1, §26.1, §26.8(vii), §5.16, Notations S.

Encodings:

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On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107 (3), pp. 253–291.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§32.12(i).

Encodings:

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The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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External Links:

ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

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38: 22.15 Inverse Functions

… ►With real variables, the solutions of the equations …The general solutions of (22.15.1), (22.15.2), (22.15.3) are, respectively, …where $m\in \mathbb{Z}$. … ►For power-series expansions see Carlson (2008).

39: 30.3 Eigenvalues

… ►These solutions exist only for eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$, $n=m,m+1,m+2,\mathrm{\dots }$, of the parameter $\lambda$. … ►has the solutions $\lambda ={\lambda }_{m+2j}^m\left({\gamma }^2\right)$, $j=0,1,2,\mathrm{\dots }$. If $p$ is an odd positive integer, then Equation (30.3.5) has the solutions $\lambda ={\lambda }_{m+2j+1}^m\left({\gamma }^2\right)$, $j=0,1,2,\mathrm{\dots }$. … ►In equation (30.3.5) we can also use … ►

§30.3(iv) Power-Series Expansion

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40: Bibliography B

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Singularities in Waves and Rays. In Les Houches Lecture Series Session XXXV, R. Balian, M. Kléman, and J.-P. Poirier (Eds.), Vol. 35, pp. 453–543.

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Referenced by:

§36.14(i).

Encodings:

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

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Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0324-7, MathReview, ZentralBlatt

Referenced by:

§26.19.

Encodings:

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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.13.

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Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Dan Polis̆evski), ZentralBlatt

Referenced by:

§2.4(iv).

Encodings:

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