Half-Period
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
22-4-2019
2189
Half-Period
An elliptic function can be characterized by its real and imaginary half-periods 
and 
(Whittaker and Watson 1990, p. 428), sometimes also denoted 
(Abramowitz and Stegun 1972, p. 630). The Wolfram Languagecommand WeierstrassHalfPeriods[![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/Half-Period/Inline4.gif" style="height:15px; width:5px" />g2, g3![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/Half-Period/Inline5.gif" style="height:15px; width:5px" />] gives the half-periods 
and 
corresponding to the invariants 
and 
for a Weierstrass elliptic function.
The notation
 |
(1)
|
is sometimes also defined (Whittaker and Watson 1990, p. 443), although Abramowitz and Stegun (1972, p. 630) instead use the definition
 |
(2)
|
In the case of a Weierstrass elliptic function, consider the modular discriminant
 |
(3)
|
If 
, then 
is real, and 
is pure imaginary. However, if 
, then 
is real and 
is pure imaginary.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 630, 1972.
Brezhnev, Y. V. "Uniformisation: On the Burnside Curve 
." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.
Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
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