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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: 5.7 Series Expansions

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§5.7(ii) Other Series

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5: 1.2 Elementary Algebra

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§1.2(iii) Partial Fractions

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6: 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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7: 19.14 Reduction of General Elliptic Integrals

… ►The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …

8: 5.19 Mathematical Applications

… ►By decomposition into partial fractions (§1.2(iii)) …

9: 10.23 Sums

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Partial Fractions

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

10: Bibliography R

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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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