This is the definition of the elliptic integral of the first 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 first kind, which is traditionally denoted by a different letter, .
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These are the partial derivatives.
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,
m
] gives the elliptic integral of the first kind 
. 
, 
. 
then 
. 
to infinity. 
on the interval 
, but the function also makes sense for more general (complex-valued) 
. 
gives the complete elliptic integral of the first kind, which is traditionally denoted by a different letter, 
. 
