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21: 27.12 Asymptotic Formulas: Primes

… ► $\pi \left(x\right)-\mathrm{li}\left(x\right)$ changes sign infinitely often as $x\to \mathrm{\infty }$; see Littlewood (1914), Bays and Hudson (2000). … ►If $a$ is relatively prime to the modulus $m$, then there are infinitely many primes congruent to $a(modm)$. … ►There are infinitely many Carmichael numbers. …

22: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►Let $q$ be a normal value (§28.12(i)) with respect to $\nu$, and $f(z)$ be a function that is analytic on a doubly-infinite open strip $S$ that contains the real axis. …

23: 1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(iii) Infinite Determinants

… ►If ${𝐷}_n[a_{j,k}]$ tends to a limit $L$ as $n\to \mathrm{\infty }$, then we say that the infinite determinant ${𝐷}_{\mathrm{\infty }}[a_{j,k}]$ converges and ${𝐷}_{\mathrm{\infty }}[a_{j,k}]=L$. ►Of importance for special functions are infinite determinants of Hill’s type. These have the property that the double series …

24: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

25: 2.3 Integrals of a Real Variable

… ►converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. … ►If, in addition, $q(t)$ is infinitely differentiable on $[0,\mathrm{\infty })$ and … ►assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. … ►Assume that $q(t)$ again has the expansion (2.3.7) and this expansion is infinitely differentiable, $q(t)$ is infinitely differentiable on $(0,\mathrm{\infty })$, and each of the integrals $\int {\mathrm{e}}^{\mathrm{i}xt}{q}^{(s)}(t)dt$, $s=0,1,2,\mathrm{\dots }$, converges at $t=\mathrm{\infty }$, uniformly for all sufficiently large $x$. … ►

On $(a,b)$, $p(t)$ and $q(t)$ are infinitely differentiable and $p^{\prime }(t)>0$.

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26: Bibliography R

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Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.

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External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii), 7th item, §10.77(ix).

Encodings:

BibTeX

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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External Links:

Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

Encodings:

BibTeX

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The problem of an infinite plate under an inclined loading, with tables of the integrals of $\mathrm{Ai}(\pm \mathrm{x})$ and $\mathrm{Bi}(\pm \mathrm{x})$ . Quart. J. Mech. Appl. Math. 7 (1), pp. 1–7.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§9.16, 1st item.

Encodings:

BibTeX

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Sources in the development of mathematics. Cambridge University Press, Cambridge.

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Notes:

Infinite Series and Products from the Fifteenth to the Twenty-first Century

External Links:

ISBN 978-0-521-11470-7, Document, Link, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

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27: 1.5 Calculus of Two or More Variables

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Infinite Integrals

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Infinite Double Integrals

►Infinite double integrals occur when $f(x,y)$ becomes infinite at points in $D$ or when $D$ is unbounded. … ►Finite and infinite integrals can be defined in a similar way. …A more general concept of integrability (both finite and infinite) for functions on domains in ${ℝ}^n$ is Lebesgue integrability. …

28: Bibliography L

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On the theory of diffraction by an aperture in an infinite plane screen. I. Phys. Rev. 74 (8), pp. 958–974.

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External Links:

Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§11.12.

Encodings:

BibTeX

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Note on a method for computing infinite integrals of oscillatory functions. Proc. Cambridge Philos. Soc. 52 (4), pp. 764–768.

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External Links:

Document, MathReview (D. J. Hofsommer), ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

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Evaluating infinite integrals involving Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 217–231.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Integrating some infinite oscillating tails. J. Comput. Appl. Math. 12/13, pp. 109–117.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

29: Bibliography O

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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A closed form solution of the $s$-wave Bethe-Goldstone equation with an infinite repulsive core. J. Math. Phys. 27 (4), pp. 1154–1158.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§30.13(v).

Encodings:

BibTeX

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30: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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5.17.2 $$G\left(n\right)=(n-2)!(n-3)!\mathrm{\cdots }1!,$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $!$: factorial (as in $n!$) and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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