Weierstrass Zeta Function
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Weierstrass Elliptic and Related Functions." Ch. 18 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
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23-4-2019
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Weierstrass Zeta Function
The Weierstrass zeta function 
is the quasiperiodic function defined by
 |
(1)
|
where 
is the Weierstrass elliptic function with invariants 
and 
, with
 |
(2)
|
As in the case of other Weierstrass elliptic functions, the elliptic invariants 
and 
are frequently suppressed for compactness. The function is implemented in the Wolfram Language as WeierstrassZeta[u, ![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/WeierstrassZetaFunction/Inline7.gif" style="height:14px; width:5px" />g2, g3![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/WeierstrassZetaFunction/Inline8.gif" style="height:14px; width:5px" />].
Using the definition above gives
where 
, so
 |
(5)
|
so 
is an odd function. Integrating 
gives
 |
(6)
|
Letting 
gives
 |
(7)
|
so
 |
(8)
|
Similarly,
 |
(9)
|
From Whittaker and Watson (1990),
 |
(10)
|
If 
, then
 |
(11)
|
(Whittaker and Watson 1990, p. 446). Also,
 |
(12)
|
(Whittaker and Watson 1990, p. 446).
The series expansion of 
is given by
 |
(13)
|
where
and
 |
(16)
|
for 
(Abramowitz and Stegun 1972, p. 635). The first few coefficients are therefore
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Weierstrass Elliptic and Related Functions." Ch. 18 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 627-671, 1972.
Brezhnev, Y. V. "Uniformisation: On the Burnside Curve 
." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.
Tölke, F. "Spezielle Weierstraßsche Zeta-Funktionen." Ch. 8 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 145-163, 1967.
Whittaker, E. T. and Watson, G. N. "Quasi-Periodic Functions. The Function 
" and "The Quasi-Periodicity of the Function 
." §20.4 and 20.41 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 445-447 and 449-451, 1990.
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