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11: 5.11 Asymptotic Expansions

… ►where

a_{0}=\tfrac{1}{2}\sqrt{2}

and …Wrench (1968) gives exact values of

g_{k}

up to

g_{20}

. … ►where

a\;(>0)

and

b\;(\in\mathbb{C})

are both fixed, and …where

h\;(\in\mathbb{C})

is fixed, and

B_{k}\left(h\right)

is the Bernoulli polynomial defined in §24.2(i). … ►In this subsection

a

b

, and

c

are real or complex constants. …

12: 7.23 Tables

… ►

§7.23(ii) Real Variables

… ►

•

Finn and Mugglestone (1965) includes the Voigt function $H\left(a,u\right)$ , $u\in[0,22]$ , $a\in[0,1]$ , 6S.

… ►

§7.23(iii) Complex Variables, $z=x+iy$

… ►

•

Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$ , $x\in[0,5]$ , $y=0.5(.5)3$ , 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$ , $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$ , $x=0(.5)20(1)25$ , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

…

13: 9.18 Tables

… ►

•

Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

… ►

§9.18(iii) Complex Variables

… ►

•

Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

►

•

Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

►

•

Zhang and Jin (1996, p. 339) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ ; 8D.

…

14: 18.40 Methods of Computation

… ►

A numerical approach to the recursion coefficients and quadrature abscissas and weights

… ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let

N

be a positive integer and define … ►Results of low (

~{}2

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

20

. … ►The quadrature points and weights can be put to a more direct and efficient use. … ►In Figure 18.40.2 the approximations were carried out with a precision of 50 decimal digits.

15: Bibliography L

… ►

R. E. Langer (1934) 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.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

… ►

D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

… ►

N. Levinson and R. M. Redheffer (1970) Complex Variables. Holden-Day Inc., San Francisco, CA.

ⓘ

External Links:: MathReview (J. E. Miller), ZentralBlatt
Referenced by:: §1.10(i), §1.10(ii), §1.10(iii), §1.10(vi), §1.10(x), §1.9(i), §1.9(ii), §1.9(iii), §1.9(iv), §1.9(vi), §4.14, §4.2(i), §4.2(iv), §4.20, §4.22, §4.23(ii), §4.23(v), §4.24(ii), §4.28, §4.31, §4.37(ii), §4.4(i), §4.4(ii), §4.7(i), §4.7(ii), §4.8(i), Ch.4.
Encodings:: BibTeX

… ►

L.-W. Li, M. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan (1998a) Computations of spheroidal harmonics with complex arguments: A review with an algorithm. Phys. Rev. E 58 (5), pp. 6792–6806.

ⓘ

Notes:: Mathematica package for eigenvalues and solutions of the spheroidal wave equations
External Links:: Code(SpheroidF), Document
Referenced by:: 2nd item, 3rd item.
Encodings:: BibTeX

…

16: 20.11 Generalizations and Analogs

… ►It is a discrete analog of theta functions. … ►Ramanujan’s theta function

f(a,b)

is defined by …where

a,b\in\mathbb{C}

and

\left|ab\right|<1

. … ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►However, in this case

q

is no longer regarded as an independent complex variable within the unit circle, because

k

is related to the variable

\tau=\tau(k)

of the theta functions via (20.9.2). …

17: 23.9 Laurent and Other Power Series

… ►

c_{2}=\frac{1}{20}g_{2},

… ►For

z\in\mathbb{C}

►

23.9.7

\sigma\left(z\right)=\sum_{m,n=0}^{\infty}a_{m,n}(10c_{2})^{m}(56c_{3})^{n}% \frac{z^{4m+6n+1}}{(4m+6n+1)!},

ⓘ

Symbols:: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $!$ : factorial (as in $n!$ ), $\mathbb{L}$ : lattice, $n$ : integer, $m$ : integer, $z$ : complex, $c_{n}$ and $a_{m,n}$ : coefficients
A&S Ref:: 18.5.6
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

►where

a_{0,0}=1

a_{m,n}=0

if either

m

n<0

, and …For

a_{m,n}

with

m=0,1,\dots,12

and

n=0,1,\dots,8

, see Abramowitz and Stegun (1964, p. 637).

18: Bibliography G

… ►

I. Gargantini and P. Henrici (1967) A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

ⓘ

External Links:: FullText, MathReview (D. C. Gilles), ZentralBlatt
Referenced by:: §10.74(v), §3.10(ii).
Encodings:: BibTeX

… ►

W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

ⓘ

Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2002c) Computing complex Airy functions by numerical quadrature. Numer. Algorithms 30 (1), pp. 11–23.

ⓘ

External Links:: ISSN 1017-1398, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §9.17(iii).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

ⓘ

Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

… ►

Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

ⓘ

External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

…

19: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in

\mathbf{x}

space, across which

\Psi_{K}(\mathbf{x};k)

\Psi^{(\mathrm{U})}(\mathbf{x};k)

acquires an exponentially-small asymptotic contribution (in

k

), associated with a complex critical point of

\Phi_{K}

\Phi^{(\mathrm{U})}

. …where

j

denotes a real critical point (36.4.1) or (36.4.2), and

\mu

denotes a critical point with complex

t

s, t

, connected with

j

by a steepest-descent path (that is, a path where

\Re\Phi=\mathrm{constant}

) in complex

t

(s,t)

space. … ►The Stokes set consists of the rays

\operatorname{ph}x=\pm 2\pi/3

in the complex

x

-plane. … ►This part of the Stokes set connects two complex saddles. … ►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. …

20: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►In this section we give asymptotic expansions of PCFs for large values of the parameter

a

that are uniform with respect to the variable

z

, when both

a

and

z

(=x)

are real. These expansions follow from Olver (1959), where detailed information is also given for complex variables. … ►Here bars do not denote complex conjugates; instead … ►(For complex values of

\mu

and

t

see Olver (1959).) … ►The variable

\zeta

is defined by …