Composition of Elliptic Integral

Theorem

An elliptic integral in $x$ can be reduced to the sum of:

an elementary function

and constant multiples of integrals in three standard forms involving $x$ and:

the (elliptic) modulus $k$

the parameter $n$.

Those three standard forms are known as Legendre's standard elliptic integrals.

Proof

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Also see

• Definition:Legendre's Standard Elliptic Integrals

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): elliptic integral
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): elliptic integral