Can constants be complex for integral table?

Question

There is a fairly well know definite integral, that appears on the tables:

$$\int^\infty_{-\infty} e^{-(ax^2 + bx+c)}~dx=\sqrt{\frac{\pi}{a}}e^{(b^2-4ac)/4a}$$

Will this be correct if $a$, $b$, or $c$ are complex? Or, putting it another way:

$$\int^\infty_{-\infty} e^{-(iax^2 + ibx)}~dx=\sqrt{\frac{\pi}{ia}}e^{ib^2/4a}$$

Is this true?

Answer 1

The formula works as long as $\Re(a) > 0$.

However, if $a$ is pure imaginary, the integral does not converge to a number, unless $b$ is pure imaginary as well. And the result is not the same: If $a$ is pure imaginary then the integral is $$ \frac{(1-i)\pi}{\sqrt{2a}}e^{(ib^2-4ac)/(4a)} $$

And if $\Re(a) < 0$ the integral always diverges.

Answer 2

In the complex case the term inside the integral is always of magnitude 1, so it never converges.

You could make the $b$ and $c$ complex without any problem.