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)du2√u
The real analysis way of evaluating this integral is to consider a parametric family:
I(ϵ)=∫∞0sin(u)2√ue−ϵudu=12∞∑n=0(−1)n(2n+1)!∫∞0u2n+12e−ϵudu=12∞∑n=0(−1)n(2n+1)!Γ(2n+32)ϵ−32−2n=12ϵ3/2∞∑n=0(−1ϵ2)nΓ(2n+32)Γ(2n+2)Γ−duplication=12ϵ3/2∞∑n=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(54,32−54)∫10x54−1(1−x)32−54−1(1+xϵ2)−3/4dx=123/2Γ(34)Γ(54)Γ(32)Γ(32)Γ(54)Γ(14)∫10x54−1(1−x)14−1(ϵ2+x)−3/4dx
Now we are ready to compute limϵ→0I(ϵ):
limϵ→0I(ϵ)=123/2Γ(34)Γ(14)∫10x12−1(1−x)14−1dx=123/2Γ(34)Γ(14)Γ(12)Γ(14)Γ(34)=123/2Γ(12)=12√π2
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