This is the definition of the elliptic integral of the second kind. Classically, the integral is defined for on the interval , but the function also makes sense for more general (complex-valued) .
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Setting gives the complete elliptic integral of the second kind, a particular case of the hypergeometric function.
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This is the indefinite integral of the complete integral .
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This is a series expansion around .
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This is the derivative of the incomplete function with respect to .
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. 
,
m
] gives the elliptic integral of the second kind 
. 
, 
. 
. 
to 
. 
to infinity. 
on the interval 
, but the function also makes sense for more general (complex-valued) 
. 
gives the complete elliptic integral of the second kind, a particular case of the hypergeometric function. 
. 
. 
. 
