representation as Weierstrass elliptic functions

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1: 23.11 Integral Representations

§23.11 Integral Representations

…

2: 23.6 Relations to Other Functions

… ►For representations of the Jacobi functions

\operatorname{sn}

\operatorname{cn}

, and

\operatorname{dn}

as quotients of

\sigma

-functions see Lawden (1989, §§6.2, 6.3). ►

§23.6(iii) General Elliptic Functions

►For representations of general elliptic functions (§23.2(iii)) in terms of

\sigma\left(z\right)

and

\wp\left(z\right)

see Lawden (1989, §§8.9, 8.10), and for expansions in terms of

\zeta\left(z\right)

see Lawden (1989, §8.11). …

3: Bibliography E

… ►

U. Eckhardt (1980) Algorithm 549: Weierstrass’ elliptic functions. ACM Trans. Math. Software 6 (1), pp. 112–120.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 18D.
External Links:: ISSN 0098-3500, Document, GAMS (module 549; package TOMS)
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

G. P. Egorychev (1984) Integral Representation and the Computation of Combinatorial Sums. Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, Providence, RI.

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Notes:: Translated from the Russian by H. H. McFadden, Translation edited by Lev J. Leifman
External Links:: ISBN 0-8218-4512-8, MathReview, ZentralBlatt
Referenced by:: §15.17(iv).
Encodings:: BibTeX

… ►

M. Eichler and D. Zagier (1982) On the zeros of the Weierstrass

\wp

-function. Math. Ann. 258 (4), pp. 399–407.

ⓘ

External Links:: ISSN 0025-5831, Document, MathReview (Toyokazu Hiramatsu), ZentralBlatt
Referenced by:: §23.13.
Encodings:: BibTeX

… ►

T. Estermann (1959) On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.

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External Links:: Document, MathReview (S. Chowla), ZentralBlatt
Referenced by:: §20.12(i).
Encodings:: BibTeX

… ►

J. A. Ewell (1990) A new series representation for

\zeta(3)

. Amer. Math. Monthly 97 (3), pp. 219–220.

ⓘ

External Links:: ISSN 0002-9890, FullText, MathReview (Christian Radoux), ZentralBlatt
Referenced by:: (25.8.10), §25.8.
Encodings:: BibTeX

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The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ has been added. In (15.6.6), the constraint $|\operatorname{ph}\left(-z\right)|<\pi$ has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when $|\operatorname{ph}\left(1-z\right)|<\pi$ , except (15.6.6) which holds for $|\operatorname{ph}\left(-z\right)|<\pi$ .”, has been removed.

… ►

Subsection 19.25(vi)

The Weierstrass lattice roots $e_{j},$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_{j}$ , and lattice invariants $g_{2}$ , $g_{3}$ , now link to their respective definitions (see §§23.2(i), 23.3(i)).

Reported by Felix Ospald.

►

Equation (19.25.37)

The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

… ►

Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

These equations were rewritten with the modulus (second argument) of the Jacobian elliptic function defined explicitly in the preceding line of text.

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Table 22.4.3

Originally a minus sign was missing in the entries for $\operatorname{cd}u$ and $\operatorname{dc}u$ in the second column (headed $z+K+iK^{\prime}$ ). The correct entries are $-k^{-1}\operatorname{ns}z$ and $-k\operatorname{sn}z$ . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\operatorname{sn},\operatorname{cn},\operatorname{dn}$ , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}K^{\prime}$	$z+\mathrm{i}K^{\prime}$	$z+2K$	$z+2K+2\mathrm{i}K^{\prime}$	$z+2\mathrm{i}K^{\prime}$
$\vdots$
$\operatorname{cd}u$	$-\operatorname{sn}z$	$-k^{-1}\operatorname{ns}z$	$k^{-1}\operatorname{dc}z$	$-\operatorname{cd}z$	$-\operatorname{cd}z$	$\operatorname{cd}z$
$\vdots$
$\operatorname{dc}u$	$-\operatorname{ns}z$	$-k\operatorname{sn}z$	$k\operatorname{cd}z$	$-\operatorname{dc}z$	$-\operatorname{dc}z$	$\operatorname{dc}z$
$\vdots$

Reported 2014-02-28 by Svante Janson.

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