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Re: question
*To*: mathgroup at smc.vnet.net
*Subject*: [mg75273] Re: [mg75255] question
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Mon, 23 Apr 2007 05:40:08 -0400 (EDT)
*References*: <200704220910.FAA20298@smc.vnet.net> <A0A84097-6960-4D4E-9BF8-64F74237A957@mimuw.edu.pl>
On 22 Apr 2007, at 23:33, Andrzej Kozlowski wrote:
> *This message was transferred with a trial version of CommuniGate
> (tm) Pro*
> On 22 Apr 2007, at 18:10, dimitris wrote:
>
>> Can somebody point me out a way 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?
>>
>
> I can't see how to do this but I can at least show that this is
> equivalent to the following curious formula for Pi:
>
> Pi == Gamma[1/4]^4/(16*EllipticK[1/2]^2)
>
> So if one can prove (or find somewhere a proof) of this formula
> then one will also obtain your derivation.
>
> Andrzej Kozlowski
I think I can now prove it. We want to show that:
Gamma[1/4]^2 == 4*EllipticF[Pi/4, 2]*Sqrt[2*Pi])
First, we shal start with some identities which Mathematica can prove.
In[1]:=
FullSimplify[EllipticK[z] == (Pi/2)*Hypergeometric2F1[1/2, 1/2, 1,
z], Abs[z] < 1]
Out[1]=
True
In[3]:=
FullSimplify[Unevaluated[Gamma[1/4]^2/(4*Sqrt[Pi]) == (Pi/2)
*Hypergeometric2F1[1/2, 1/2, 1, 1/2]]]
Out[3]=
True
This leads us to
Gamma[1/4]^2/(4*Sqrt[Pi])==EllipticK[1/2]
which is what I wrote above, was equivalent to the result you wanted
proved. Let us demonstrate that. First, by definition of EllipticK
EllipticK[1/2]==EllipticF[Pi/2,1/2]
Now we refer to page 593 of Abramowitz and Stegun:
EllipticF[theta, m] == Sqrt[m]*EllipticF[phi, m^(-1)]
where Sin[phi] == m^(1/2)*Sin[theta]
this gives us
EllipticK[1/2]==EllipticF[Pi/2,1/2]==Sqrt[2] * EllipticF[Pi/4,2]
(Mathematica can't prove the last relationship because it always
reduced EllipticF[Pi/2,a] to EllipticK[a] but you can do:
N[EllipticF[Pi/2, 1/2] - Sqrt[2]*EllipticF[Pi/4, 2], 100] // Chop)
which gives us:
Gamma[1/4]^2/(4*Sqrt[Pi]) ==Sqrt[2] * EllipticF[Pi/4,2]
which is what you wanted to prove.
Andrzej Kozlowski
**References**:
**question**
*From:* dimitris <dimmechan@yahoo.com>
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