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21: Bibliography L

… ►

The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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External Links:

ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. Constr. Approx. 17 (1), pp. 59–90.

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External Links:

ISSN 0176-4276, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

… ►

Asymptotics of the first Appell function $F_1$ with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

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External Links:

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

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

… ►

Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19, pp. 215–250.

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External Links:

Document, MathReview (Colin Clark), ZentralBlatt

Referenced by:

§36.12(iii), §36.14(i).

Encodings:

BibTeX

… ►

Miniaturized tables of Bessel functions. III. Math. Comp. 26 (117), pp. 237–240 and A14–B5.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

BibTeX

…

22: Staff

… ►

Ronald F. Boisvert, Editor at Large, NIST

… ►

Frank W. J. Olver, University of Maryland and NIST, Chaps. 1, 2, 4, 9, 10

►

Richard B. Paris, University of Abertay, Chaps. 8, 11

… ►

Diego Dominici, State University of New York at New Paltz, for Chaps. 9, 10 (deceased)

… ►

Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11 (deceased)

…

23: 15.13 Zeros

… ►Let $N(a,b,c)$ denote the number of zeros of $F(a,b;c;z)$ in the sector . … ►If $a$, $b$, $c$, $c-a$, or $c-b\in \{0,-1,-2,\mathrm{\dots }\}$, then $F(a,b;c;z)$ is not defined, or reduces to a polynomial, or reduces to ${(1-z)}^{c-a-b}$ times a polynomial. …

24: 2.3 Integrals of a Real Variable

… ►Without loss of generality, we assume that this minimum is at the left endpoint $a$. … ►For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. … ►it is free from singularity at $t=\alpha$. … ► $\kappa =\kappa (\alpha )$ being the value of $w$ at $t=k$. …with the coefficients ${\varphi }_s(\alpha )$ continuous at $\alpha =0$. …

25: 2.4 Contour Integrals

… ►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty })$. … ►If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i)) … ►Now suppose that in (2.4.10) the minimum of $\mathrm{\Re }\left(zp(t)\right)$ on $𝒫$ occurs at an interior point $t_0$. … ►Cases in which $p^{\prime }(t_0)\ne 0$ are usually handled by deforming the integration path in such a way that the minimum of $\mathrm{\Re }\left(zp(t)\right)$ is attained at a saddle point or at an endpoint. … ►with $p,q$ and their derivatives evaluated at $t_0$. …

26: 31.3 Basic Solutions

… ►

§31.3(i) Fuchs–Frobenius Solutions at $z=0$

… ►In general, one of them has a logarithmic singularity at $z=0$. ►

§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

… ►Solutions of (31.2.1) corresponding to the exponents $0$ and $1-\delta$ at $z=1$ are respectively, … ►Solutions of (31.2.1) corresponding to the exponents $0$ and $1-ϵ$ at $z=a$ are respectively, …

27: Frank W. J. Olver

… ►In 1945–1961 he was a founding member of the Mathematics Division and Head of the Numerical Methods Section at the National Physical Laboratory, Teddington, U. … ►Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions. … ►After leaving NIST in 1986, Olver became a professor at the Institute for Physical Science and Technology and the Department of Mathematics at the University of Maryland. … ►He also spent time as a Visiting Fellow, or Professor, at the University of Lancaster, U. … ►He continued his editing work until the time of his death on April 22, 2013 at age 88.

28: 9.19 Approximations

… ►

Németh (1992, Chapter 8) covers $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, and integrals ${\int }_0^x\mathrm{Ai}\left(t\right)dt$, ${\int }_0^x\mathrm{Bi}\left(t\right)dt$, ${\int }_0^x{\int }_0^v\mathrm{Ai}\left(t\right)dtdv$, ${\int }_0^x{\int }_0^v\mathrm{Bi}\left(t\right)dtdv$ (see also (9.10.20) and (9.10.21)). The Chebyshev coefficients are given to 15D. Chebyshev coefficients are also given for expansions of the second and higher (real) zeros of $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, again to 15D.

… ►

Corless et al. (1992) describe a method of approximation based on subdividing $ℂ$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$.

…

29: 31.11 Expansions in Series of Hypergeometric Functions

… ►The Fuchs-Frobenius solutions at $\mathrm{\infty }$ are ►

31.11.3_1 $$P_j^5=\frac{{\left(\lambda \right)}_j{\left(1-\gamma +\lambda \right)}_j}{{\left(1+\lambda -\mu \right)}_{2j}}{z}^{-\lambda -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\lambda +j,1-\gamma +\lambda +j}{1+\lambda -\mu +2j};\frac{1}{z}),$$

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Defines:

$P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

… ►Then the Fuchs–Frobenius solution at $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with …and (31.11.1) converges to (31.3.10) outside the ellipse $ℰ$ in the $z$-plane with foci at 0, 1, and passing through the third finite singularity at $z=a$. … ►For example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha$ at $\mathrm{\infty }$. …

30: 13.11 Series

… ►

13.11.2 $$M(a,b,z)=\mathrm{\Gamma }\left(b-a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{a-b+\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2b-2a-1\right)}_s{\left(b-2a\right)}_s(b-a-\frac{1}{2}+s)}{{\left(b\right)}_{s}s!}{I}_{b-a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $b-a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Proof sketch:

Combine (13.24.1) with (13.14.4).

A&S Ref:

13.3.6

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

… ►

13.11.3 $$𝐌(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}\sum _{s=0}^{\mathrm{\infty }}{A}_s{\left(b-2a\right)}^{\frac{1}{2}(1-b-s)}{\left(\frac{1}{2}z\right)}^{\frac{1}{2}(1-b+s)}{J}_{b-1+s}\left(\sqrt{2z(b-2a)}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer, $z$: complex variable and $A_n$: coefficients

Source:

Slater (1960, §3.8)

A&S Ref:

13.3.7

Referenced by:

§13.11, Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

… ►

$A_0=1,$

►

$A_1=0,$

►

$A_2={\displaystyle \frac{1}{2}}b,$

…