expansions in doubly-infinite partial fractions

SearchAdvancedHelp

(0.003 seconds)

31—40 of 930 matching pages

31: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

►

§10.40(i) Hankel’s Expansions

… ► … ►In the expansion (10.40.2) assume that $z>0$ and the sum is truncated when $k=\mathrm{\ell }-1$. … ►

§10.40(iv) Exponentially-Improved Expansions

…

32: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

33: Bibliography T

… ►

Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.

ⓘ

External Links:

ISSN 0020-2932, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.8(ii), §13.8(ii).

Encodings:

BibTeX

… ►

The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

… ►

Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§13.8(iii), §13.8(iii).

Encodings:

BibTeX

… ►

Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

ⓘ

External Links:

MathReview (H. Pollard)

Referenced by:

§1.18(x).

Encodings:

BibTeX

►

Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

ⓘ

External Links:

ISBN 0198533187, MathReview Entry

Referenced by:

§1.18(x).

Encodings:

BibTeX

…

34: 1.10 Functions of a Complex Variable

… ►Note that (1.10.4) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6). … ►(In other words $n{F}_n$ is the coefficient of ${(z-z_0)}^{-1}$ in the Laurent expansion of $1/{(f(z)-f(z_0))}^n$ in powers of $(z-z_0)$; compare §1.10(iii).) … ►

§1.10(x) Infinite Partial Fractions

… ►

Mittag-Leffler’s Expansion

… ►Let $F(x,z)$ have a converging power series expansion of the form …

35: 29.20 Methods of Computation

… ►The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … ►The numerical computations described in Jansen (1977) are based in part upon this method. … ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …

36: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). These expansions are uniform with respect to $z$, including the turning point $z_0$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … ►If $f(z)$ has a double zero $z_0$, or more generally $z_0$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots }$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. … ►In regions in which the function $f(z)$ has a simple pole at $z=z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z=z_0$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho }$, where $\rho$ is the limiting value of ${(z-z_0)}^{2}g(z)$ as $z\to z_0$. …

37: 33.20 Expansions for Small $|ϵ|$

… ►

§33.20(i) Case $ϵ=0$

… ►

§33.20(ii) Power-Series in $ϵ$ for the Regular Solution

… ►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and … ►

§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

38: 10.23 Sums

… ►

Partial Fractions

►For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005). … ►

§10.23(iii) Series Expansions of Arbitrary Functions

… ►

Other Series Expansions

… ► …

39: Bibliography O

… ►

Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.8(v), §22.11, §4.27.

Encodings:

BibTeX

… ►

Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

BibTeX

… ►

On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

… ►

Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.

ⓘ

External Links:

Document, MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

… ►

Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii), §2.7(ii).

Encodings:

BibTeX

…

40: 28.15 Expansions for Small $q$

§28.15 Expansions for Small $q$

►

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

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

28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

ⓘ

Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

pMML, png, TeX

See also:

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

►

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

…