expansions of solutions in series of

(0.010 seconds)

21—30 of 62 matching pages

21: 2.8 Differential Equations with a Parameter

… ►The form of the asymptotic expansion depends on the nature of the transition points in

\mathbf{D}

, that is, points at which

f(z)

has a zero or singularity. … ►In Cases I and II the asymptotic solutions are in terms of the functions that satisfy (2.8.8) with

\psi(\xi)=0

. … ►For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004). … ►

§2.8(iv) Case III: Simple Pole

… ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). …

22: 4.13 Lambert $W$ -Function

… ►The Lambert

W

-function

W\left(z\right)

is the solution of the equation … ►On the

z

-interval

[0,\infty)

there is one real solution, and it is nonnegative and increasing. … ►Other solutions of (4.13.1) are other branches of

W\left(z\right)

. … ►In the case of

k=0

and real

z

the series converges for

z\geq\mathrm{e}

. …For these results and other asymptotic expansions see Corless et al. (1997). …

23: 3.6 Linear Difference Equations

… ►This is of little consequence if the wanted solution is growing in magnitude at least as fast as any other solution of (3.6.3), and the recursion process is stable. … ►Because the recessive solution of a homogeneous equation is the fastest growing solution in the backward direction, it occurred to J. …A “trial solution” is then computed by backward recursion, in the course of which the original components of the unwanted solution

g_{n}

die away. … ►If, as

n\to\infty

, the wanted solution

w_{n}

grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. … ►Here

\ell\in[0,k]

, and its actual value depends on the asymptotic behavior of the wanted solution in relation to those of the other solutions. …

24: Bibliography K

… ►

D. Karp, A. Savenkova, and S. M. Sitnik (2007) Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt
Referenced by:: §19.12, §19.5.
Encodings:: BibTeX

… ►

A. V. Kashevarov (1998) The second Painlevé equation in electric probe theory. Some numerical solutions. Zh. Vychisl. Mat. Mat. Fiz. 38 (6), pp. 992–1000 (Russian).

ⓘ

Notes:: In Russian. English translation: Comput. Math. Math. Phys. 38(1998), no. 6, pp. 950–958
External Links:: ISSN 0044-4669, MathReview, ZentralBlatt
Referenced by:: §32.17.
Encodings:: BibTeX

►

A. V. Kashevarov (2004) The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Technical Physics 49 (1), pp. 1–7.

ⓘ

External Links:: Document
Referenced by:: §32.17.
Encodings:: BibTeX

… ►

M. Katsurada (2003) Asymptotic expansions of certain

q

-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

ⓘ

External Links:: ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §17.2(i).
Encodings:: BibTeX

… ►

C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively

q

-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

ⓘ

Notes:: Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.
External Links:: Code, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

…

25: 9.17 Methods of Computation

… ►

§9.17(i) Maclaurin Expansions

►Although the Maclaurin-series expansions of §§9.4 and 9.12(vi) converge for all finite values of

z

, they are cumbersome to use when

|z|

is large owing to slowness of convergence and cancellation. For large

|z|

the asymptotic expansions of §§9.7 and 9.12(viii) should be used instead. …However, in the case of

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

and

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

this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … ►As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. …

26: Bibliography S

… ►

M. J. Seaton (2002b) FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

ⓘ

Notes:: Single-precision Fortran code.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 10th item.
Encodings:: BibTeX

… ►

H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §12.13(i).
Encodings:: BibTeX

►

H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

ⓘ

External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

… ►

B. D. Sleeman (1966b) The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.

ⓘ

External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.17(i).
Encodings:: BibTeX

… ►

S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

ⓘ

External Links:: ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt
Referenced by:: §17.3(i).
Encodings:: BibTeX

…

27: 12.4 Power-Series Expansions

§12.4 Power-Series Expansions

… ►where the initial values are given by (12.2.6)–(12.2.9), and

u_{1}(a,z)

and

u_{2}(a,z)

are the even and odd solutions of (12.2.2) given by ►

12.4.3

u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(1+(a+\tfrac{1}{2})\frac{z^{2}}{2!}+(a+% \tfrac{1}{2})(a+\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),

ⓘ

Defines:: $u_{1}(a,z)$ : solution (locally)
Symbols:: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.2.1 (modification of)
Referenced by:: §12.14(v), §12.7(iv)
Permalink:: http://dlmf.nist.gov/12.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

►

12.4.4

u_{2}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(z+(a+\tfrac{3}{2})\frac{z^{3}}{3!}+(a+% \tfrac{3}{2})(a+\tfrac{7}{2})\frac{z^{5}}{5!}+\cdots\right).

ⓘ

Defines:: $u_{2}(a,z)$ : solution (locally)
Symbols:: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.2.3 (modification of)
Permalink:: http://dlmf.nist.gov/12.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

… ►These series converge for all values of

z

28: 11.13 Methods of Computation

… ►

§11.13(ii) Series Expansions

►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of

z

, they are cumbersome to use when

|z|

is large owing to slowness of convergence and cancellation. For large

|z|

and/or

|\nu|

the asymptotic expansions given in §11.6 should be used instead. … ►To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. … ►The solution

\mathbf{K}_{\nu}\left(x\right)

needs to be integrated backwards for small

x

, and either forwards or backwards for large

x

depending whether or not

\nu

exceeds

\tfrac{1}{2}

. …

29: 13.29 Methods of Computation

… ►

§13.29(i) Series Expansions

►Although the Maclaurin series expansion (13.2.2) converges for all finite values of

z

, it is cumbersome to use when

|z|

is large owing to slowness of convergence and cancellation. …However, this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied by the combination of (13.7.10) and (13.7.11), or by use of the hyperasymptotic expansions given in Olde Daalhuis and Olver (1995a). … ►As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. … ►with recessive solution …

30: 12.14 The Function $W\left(a,x\right)$

… ►In this section solutions of equation (12.2.3) are considered. … ►

§12.14(v) Power-Series Expansions

… ►

Airy-type Uniform Expansions

… ►In this case there are no real turning points, and the solutions of (12.2.3), with

z

replaced by

x

, oscillate on the entire real

x

-axis. … ►As noted in §12.14(ix), when

a

is negative the solutions of (12.2.3), with

z