Complete Elliptic Integral of the First Kind
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.
الجزء والصفحة:
...
25-4-2019
3804
Complete Elliptic Integral of the First Kind
The complete elliptic integral of the first kind 
, illustrated above as a function of the elliptic modulus 
, is defined by
where 
is the incomplete elliptic integral of the first kind and 
is the hypergeometric function.
It is implemented in the Wolfram Language as EllipticK[m], where 
is the parameter.
It satisfies the identity
 |
(4)
|
where 
is a Legendre polynomial. This simplifies to
 |
(5)
|
for all complex values of 
except possibly for real 
with 
.
In addition, 
satisfies the identity
 |
(6)
|
where 
is the complementary modulus. Amazingly, this reduces to the beautiful form
 |
(7)
|
for 
(Watson 1908, 1939).
can be computed in closed form for special values of 
, where 
is a called an elliptic integral singular value. Other special values include
satisfies
 |
(13)
|
possibly modulo issues of 
, which can be derived from equation 17.4.17 in Abramowitz and Stegun (1972, p. 593).
is related to the Jacobi elliptic functions through
 |
(14)
|
where the nome is defined by
 |
(15)
|
with 
, where 
is the complementary modulus.
satisfies the Legendre relation
 |
(16)
|
where 
and 
are complete elliptic integrals of the first and second kinds, respectively, and 
and 
are the complementary integrals. The modulus 
is often suppressed for conciseness, so that 
and 
are often simply written 
and 
, respectively.
The integral 
for complementary modulus is given by
 |
(17)
|
(Whittaker and Watson 1990, p. 501), and
(Whittaker and Watson 1990, p. 521), so
(cf. Whittaker and Watson 1990, p. 521).
The solution to the differential equation
![d/(dk)[k(1-k^2)(dy)/(dk)]-ky=0](http://mathworld.wolfram.com/images/equations/CompleteEllipticIntegraloftheFirstKind/NumberedEquation10.gif) |
(22)
|
(Zwillinger 1997, p. 122; Gradshteyn and Ryzhik 2000, p. 907) is
 |
(23)
|
where the two solutions are illustrated above and 
.
Definite integrals of 
include
where 
(not to be confused with 
) is Catalan's constant.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.
Watson G. N. "The Expansion of Products of Hypergeometric Functions." Quart. J. Pure Appl. Math. 39, 27-51, 1907.
Watson G. N. "A Series for the Square of the Hypergeometric Function." Quart. J. Pure Appl. Math. 40, 46-57, 1908.
Watson, G. N. "Three Triple Integrals." Quart. J. Math., Oxford Ser. 2 10, 266-276, 1939.
Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997.
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