Complete Elliptic Integral of the Second Kind
المؤلف:
Gradshteyn, I. S. and Ryzhik, I. M
المصدر:
Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.
الجزء والصفحة:
...
18-8-2018
2511
Complete Elliptic Integral of the Second Kind
The complete elliptic integral of the second kind, illustrated above as a function of 
, is defined by
where 
is an incomplete elliptic integral of the second kind, 
is the hypergeometric function, and 
is a Jacobi elliptic function.
It is implemented in the Wolfram Language as EllipticE[m], where 
is the parameter.
can be computed in closed form in terms of 
and the elliptic alpha function 
for special values of 
, where 
is a called an elliptic integral singular value. Other special values include
The complete elliptic integral of the second kind satisfies the Legendre relation
 |
(7)
|
where 
and 
are complete elliptic integrals of the first and second kinds, respectively, and 
and 
are the complementary integrals. The derivative is
 |
(8)
|
(Whittaker and Watson 1990, p. 521).
The solution to the differential equation
 |
(9)
|
(Zwillinger 1997, p. 122; Gradshteyn and Ryzhik 2000, p. 907) is given by
 |
(10)
|
If 
is a singular value (i.e.,
 |
(11)
|
where 
is the elliptic lambda function), and 
and the elliptic alpha function 
are also known, then
![E(k)=(K(k))/(sqrt(r))[pi/(3[K(k)]^2)-alpha(r)]+K(k).](http://mathworld.wolfram.com/images/equations/CompleteEllipticIntegraloftheSecondKind/NumberedEquation6.gif) |
(12)
|
REFERENCES:
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.
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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