infinite partial fractions

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1: 4.36 Infinite Products and Partial Fractions

§4.36 Infinite Products and Partial Fractions

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2: 4.22 Infinite Products and Partial Fractions

§4.22 Infinite Products and Partial Fractions

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3: 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

4: 1.10 Functions of a Complex Variable

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§1.10(x) Infinite Partial Fractions

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Mittag-Leffler’s Expansion

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5: 1.2 Elementary Algebra

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§1.2(iii) Partial Fractions

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6: 5.19 Mathematical Applications

… ►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. … ►By decomposition into partial fractions (§1.2(iii)) … ►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …

7: Bibliography R

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A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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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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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

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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8: 2.4 Contour Integrals

… ►in which $a$ is finite, $b$ is finite or infinite, and $\omega$ is the angle of slope of $𝒫$ at $a$, that is, $lim(\mathrm{ph}\left(t-a\right))$ as $t\to a$ along $𝒫$. … ►However, if $p^{\prime }(t_0)=0$, then $\mu \ge 2$ and different branches of some of the fractional powers of $p_0$ are used for the coefficients $b_s$; again see §2.3(iii). … ►Suppose that on the integration path $𝒫$ there are two simple zeros of $\partial p(\alpha ,t)/\partial t$ that coincide for a certain value $\widehat{\alpha }$ of $\alpha$. … ►with $a$ and $b$ chosen so that the zeros of $\partial p(\alpha ,t)/\partial t$ correspond to the zeros $w_1(\alpha ),w_2(\alpha )$, say, of the quadratic $w^2+2aw+b$. … ►

2.4.20 $$f(\alpha ,w)=q(\alpha ,t)\frac{dt}{dw}=q(\alpha ,t)\frac{w^2+2aw+b}{\partial p(\alpha ,t)/\partial t}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $p(t)$: analytic function, $q(t)$: analytic function, $w$: change of variable, $a$ , $b$ and $f(\alpha ,w)$: function

Permalink:

http://dlmf.nist.gov/2.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(v), §2.4 and Ch.2

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