question

Hello.

In Mathematica 5.2 I got

In[73]:=
Integrate[1/Sqrt[Sin[u]], {u, 0, Pi/2}]

Out[73]=
2*EllipticF[Pi/4, 2]

The result ic correct as a numerical integration indicates.

In[74]:=
{Chop[N[%]], NIntegrate[1/Sqrt[Sin[u]], {u, 0, Pi/2}]}

Out[74]=
{2.6220575244897986, 2.622057554312378}

It can be proved that the definite integral under discussion is
given by Gamma[1/4]^2/(2*Sqrt[2*Pi]).

In[75]:=
N[Gamma[1/4]]^2/(2*Sqrt[2*Pi])

Out[75]=
2.6220575542921196

Can somebody point me out a way (by hand or a CAS) to show that the
Gamma[1/4]^2/(2*Sqrt[2*Pi]) is equal to 2*EllipticF[Pi/4, 2] or/and a
series
of steps from taking from the latter to the former and vice versa?

Thanks
Dimitris

.

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