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Hello!
I am doing this purely out of curiousity.
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
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