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Some-Values-n-e-pin-2-pi-1-4-3-4-n-e-2pin-2-pi-1-4-3-4-6-4-2-1-4-2-n-e-6pin-2-pi-1-4-3-4-1-1-4-3-1-




Question Number 127982 by Dwaipayan Shikari last updated on 03/Jan/21
Some Values ..  Σ_(n=−∞) ^∞ e^(−πn^2 ) =(π^(1/4) /(Γ((3/4))))  Σ_(n=−∞) ^∞ e^(−2πn^2 ) =(π^(1/4) /(Γ((3/4)))) (((6+4(√2)))^(1/4) /2)  Σ_(n=−∞) ^∞ e^(−6πn^2 ) =(π^(1/4) /(Γ((3/4)))).((√((1)^(1/4) +(3)^(1/4) +(4)^(1/4) +(9)^(1/4) ))/( (√(1728))))  Any Idea to prove ?
SomeValues..n=eπn2=π14Γ(34)n=e2πn2=π14Γ(34)6+4242n=e6πn2=π14Γ(34).14+34+44+941728AnyIdeatoprove?
Commented by Dwaipayan Shikari last updated on 03/Jan/21
There are so many theta function values derived by Ramanujan  Σ_(n=−∞) ^∞ e^(−3πn^2 ) =((π)^(1/4) /(Γ((3/4)))).(((27+18(√3)))^(1/4) /3)  Σ_(n=−∞) ^∞ e^(−4πn^2 ) =((π)^(1/4) /(Γ((3/4)))).(((8)^(1/4) +2)/4)  Σ_(n=−∞) ^∞ e^(−7πn^2 ) =((π)^(1/4) /(Γ((3/4)))).((√(7+4(√7)+5((28))^(1/4) +((1378))^(1/4) ))/( (√7)))
TherearesomanythetafunctionvaluesderivedbyRamanujann=e3πn2=π4Γ(34).27+18343n=e4πn2=π4Γ(34).84+24n=e7πn2=π4Γ(34).7+47+5284+137847

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