Say I have an integral such as:
$$ \int_{a}^{b} f(x) dx$$
Is it too naive, or is it valid to use a substitution which would bring the domain into the complex plane? For example, if I used the substitution $y=ix$ or $y=(5+4i)x$.

Question

Say I have an integral such as:
$$ \int_{a}^{b} f(x) dx$$
Is it too naive, or is it valid to use a substitution which would bring the domain into the complex plane? For example, if I used the substitution $y=ix$ or $y=(5+4i)x$.

daniel.ohearn1 · Answer

Seems like something that could work as a visual reference.  Why do you ask?

bobo-i-bo · Answer

@daniel.ohearn1
What do you mean by "could work as a visual reference"?

There was an integration question on OS which I was thinking, using a substitution of $y=ix$ could possibly get somewhere. :P

bobo-i-bo · Answer

@kainui

thomas5267 · Answer

I don't know.  The basic version of fundamental theorem of calculus only works for real functions.  However I think this substitution should work for some suitable functions but I don't have a proof.

daniel.ohearn1 · Answer

i would be treated like a constant.  x would have a real part and a complex part.

daniel.ohearn1 · Answer

Something a bit more interesting is the Integral of i^i^x dx