What is the Integral of exp(-i*x^2) from -infinity to +infinity?

[VVS]

Hello!

I am doing this purely out of curiousity.

Homework Statement

I am trying to integrate exp(-i*x^2) from -infinity to +infinity. Given that the integral from -infinity to infinity of exp(-x^2)=sqrt(pi).

I typed it in Wolfram Alpha and I got (1/sqrt(2)-i*1/sqrt(2))*sqrt(pi).

One can arrive at this solution by substituting the integral y=sqrt(i)x

Then one gets (1/sqrt(2)-i/sqrt(2))*integral exp(-y^2) from -infinity to infinity

BUT here is the catch. The limits have changed from -infinity to infinity to -infinity-i*infinity to +infinity+i*infinity.

However if you just evaluate the integral from -infinity to +infinity you get the right answer.
How can it be right? Isn't it mathematically inprecise? Or is there a mathematical theorem in complex analysis in which -infinity is the same the -i*infinity or something like that?

I am really curious to know.

Thank you

 

[D H]

Expand $\exp(-ix^2)$ and you get $\cos(x^2) - i\sin(x^2)$. Both $\int_{-\infty}^{\infty} \cos(x^2)\,dx$ and $\int_{-\infty}^{\infty} \sin(x^2)\,dx$ are well-defined. Google "Fresnel integral" for more info.