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Re: question

Offhand I'm not sure how to do this with Mathematica, but 

2*EllipticF[Pi/4, 2] == Sqrt[2]*EllipticK[1/2], and by

Sqrt[2]*EllipticK[1/2] == Sqrt[2]*8*Pi^(3/2)/Gamma[-1/4]^2

After this, of course, you can use a reflection formula.

Wolfram Research

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