The complete elliptic integral of the second kind, illustrated above as a function of 
, is defined by
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 |  | ![pi/2{1-sum_(n=1)^(infty)[((2n-1)!!)/((2n)!!)]^2(k^(2n))/(2n-1)}](/images/equations/CompleteEllipticIntegraloftheSecondKind/Inline7.svg) |
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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
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The complete elliptic integral of the second kind satisfies the Legendre relation
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where 
and 
are complete elliptic
integrals of the first and second kinds, respectively, and 
and 
are the complementary integrals. The derivative
is
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(Whittaker and Watson 1990, p. 521).
The solution to the differential equation
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(Zwillinger 1997, p. 122; Gradshteyn and Ryzhik 2000, p. 907) is given by
![y=C_1E(k)+C_2[E^'(k)-K^'(k)].](/images/equations/CompleteEllipticIntegraloftheSecondKind/NumberedEquation4.svg) |
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If 
is a singular value (i.e.,
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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).](/images/equations/CompleteEllipticIntegraloftheSecondKind/NumberedEquation6.svg) |
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