I was playing around with the Fresnel integrals and I've come up with a proof for the fact ∫∞0cosx2dx=∫∞0sinx2dx=√π8
The proof goes as follows
Use the fact that ∫∞−∞e−u2du=√π
Letting u=√−i⋅x,du=√−i∗dx
√−i⋅∫∞−∞eix2du=√π
∫∞−∞eix2dx=∫∞−∞cosx2+i⋅∫∞−∞sinx2dx=√−πi=(1+i)√π2
Equating real and imaginary parts (we can do this because we can assume that the two integrals have real values), we get
∫∞−∞cosx2dx=∫∞−∞sinx2dx=√π2
Then, using the fact that cosx2 and sinx2 are even functions, we get the fact above.
My question stems from the substitution made. After the substitution is made, the bounds of the integral change from ±∞ to ±(1+i)∞
Is this integral valid? Is it possible to have a complex number in the bounds of the integral?
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