divergent integrals

SearchAdvancedHelp

(0.001 seconds)

11—16 of 16 matching pages

11: 2.1 Definitions and Elementary Properties

… ►

2.1.11 $${\int }_x^{\mathrm{\infty }}f(t)dt\sim -\frac{x^{\nu +1}}{\nu +1},$$ ,

ⓘ

Symbols:

$\sim$: asymptotic equality, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $f(x)$: continuous function and $\nu$: complex constant

Permalink:

http://dlmf.nist.gov/2.1.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.1(ii), §2.1 and Ch.2

►

2.1.12

ⓘ

Symbols:

$\sim$: asymptotic equality, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $f(x)$: continuous function and $\nu$: complex constant

Permalink:

http://dlmf.nist.gov/2.1.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.1(ii), §2.1 and Ch.2

… ►Let $\sum a_s{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, …

12: 9.17 Methods of Computation

… ►Since these expansions diverge, the accuracy they yield is limited by the magnitude of $|z|$. … ►

§9.17(iii) Integral Representations

►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. … ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …

13: Bibliography W

… ►

Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ and related integrals. Optica Acta 14 (3), pp. 317–322.

ⓘ

External Links:

Document, MathReview (S. D. Bajpai)

Referenced by:

§6.14(iii), §6.7(iv).

Encodings:

BibTeX

… ►

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.

ⓘ

External Links:

Document

Referenced by:

§2.11(vi), §3.9(v), §3.9(vi).

Encodings:

BibTeX

►

Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

ⓘ

External Links:

Document

Referenced by:

§2.11(vi), §2.11(vi).

Encodings:

BibTeX

… ►

Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§10.22(vi), §10.23(iv), §10.43(vi).

Encodings:

BibTeX

… ►

A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

ⓘ

External Links:

Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§13.23(iv).

Encodings:

BibTeX

…

14: 1.9 Calculus of a Complex Variable

… ►

Poisson Integral

… ►

§1.9(v) Infinite Sequences and Series

… ►The series is divergent if $s_n$ does not converge. …A series ${\sum }_{n=0}^{\mathrm{\infty }}{z}_n$ converges (diverges) absolutely when ($>1$), or when ($>1$). … ►If the limit exists, then the double series is convergent; otherwise it is divergent. …

15: Bibliography F

… ►

Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

ⓘ

External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

… ►

On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

ⓘ

External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

… ►

Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

ⓘ

Notes:

Reprint with corrections of two books published originally in 1916 and 1936.

External Links:

MathReview

Referenced by:

§2.10(iii).

Encodings:

BibTeX

… ►

Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.36(iv).

Encodings:

BibTeX

… ►

Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§20.7(ii), §20.7(ii).

Encodings:

BibTeX

…

16: 16.2 Definition and Analytic Properties

… ►On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\mathrm{\Re }{\gamma }_q>0$, convergent except at $z=1$ if , and divergent if $\mathrm{\Re }{\gamma }_q\le -1$, where … ►In general the series (16.2.1) diverges for all nonzero values of $z$. … ►See §16.5 for the definition of ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ as a contour integral when $p>q+1$ and none of the $a_k$ is a nonpositive integer. …