A closed form evaluation of the elliptic integral
A closed form evaluation of the elliptic integral
The complete elliptic integral of the first kind <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is evaluated in terms of gamma functions for the moduli <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals square-root 2 minus …