Wednesday, April 24, 2019

integration - Evaluating inti0nftysinx2,dx with real methods?



I have seen the Fresnel integral



0sinx2dx=π8



evaluated by contour integration and other complex analysis methods, and I have found these methods to be the standard way to evaluate this integral. I was wondering, however, does anyone know a real analysis method to evaluate this integral?


Answer




Let u=x2, then
0sin(u)du2u


The real analysis way of evaluating this integral is to consider a parametric family:
I(ϵ)=0sin(u)2ueϵudu=12n=0(1)n(2n+1)!0u2n+12eϵudu=12n=0(1)n(2n+1)!Γ(2n+32)ϵ322n=12ϵ3/2n=0(1ϵ2)nΓ(2n+32)Γ(2n+2)Γduplication=12ϵ3/2n=0(1ϵ2)nΓ(n+34)Γ(n+54)2n!Γ(n+32)=1(2ϵ)3/2Γ(34)Γ(54)Γ(32)2F1(34,54;32;1ϵ2)Euler integral=1(2ϵ)3/2Γ(34)Γ(54)Γ(32)1B(54,3254)10x541(1x)32541(1+xϵ2)3/4dx=123/2Γ(34)Γ(54)Γ(32)Γ(32)Γ(54)Γ(14)10x541(1x)141(ϵ2+x)3/4dx

Now we are ready to compute limϵ0I(ϵ):
limϵ0I(ϵ)=123/2Γ(34)Γ(14)10x121(1x)141dx=123/2Γ(34)Γ(14)Γ(12)Γ(14)Γ(34)=123/2Γ(12)=12π2


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