Wednesday, June 22, 2016

calculus - Relation between integral, gamma function, elliptic integral, and AGM

The integral 0dx1+x4 is equal to Γ(14)24π.



It is calculated or verified with a computer algebra system that Γ(14)24π=K(12) , where K(m) is the complete elliptic integral of the first kind. This is in relation to what is called the elliptic integral singular value.




It is also known or verified that
K(12)=π2011sin2(t)2dt=12π201sin(t)cos(t)dt.



Can one prove directly or analytically that



0dx1+x4=12π201sin(t)cos(t)dt=π2011sin2(t)2dt=K(12) ?

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