Wednesday, 25 April 2018

calculus - Strange Limit in Proof of the Fresnel Integral

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 eu2du=π
Letting u=ix,du=idx

ieix2du=π
eix2dx=cosx2+isinx2dx=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?

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...