# Complex integral - should we use contour integration or substitution?

My question is: how is the following integral calculated: $$\int_{0 }^{\infty} e^{x(1+it)} dx$$

Here, $$i$$ is imaginary unit, and $$t \in \mathbb{R}$$ is some parameter. My questions are:

1. Can we use complex substitution e.g. $$u= x(1- it )$$ and why? If we can do it, how do we determine limits of integral?

2. If we cannot use complex substitution, how do we determine contour for integration? I know how Residuum theorem is applied, but I don't know which contour to take and also.

3. I am confused when it is allowed to use substitution and when to use contour integration.

Also, I would be very grateful if someone can recommend any book related to Complex integration (with lot of exercises and explanations, for beginners). Thanks in advance.

• If I recall correctly complex-valued substitutions can be pretty dodgy, however here I think that you could use it as a tool to find an antiderivative for your integrand and then use the Fundamental Theorem of Contour Integration Commented Dec 5, 2021 at 10:45
• The integral does not exist; you cannot evaluate it by any method. Commented Dec 5, 2021 at 11:41

There is no reason to use substitution or contour inegration. Since you cannot show that the integral converges - using any contour that goes to infinity (say along the imaginary axis) will not help the case. You can simply do the real and imaginary part of the indefinite integral: $$\int e^x\cos tx-i\int e^x\sin tx$$ Each of this integrals can be done by integrating by parts twice and moving the resulting integral to the left. We consider: $$\lim_{x\to\infty}[ I(x)] -I(0)=I(\infty)-I(0)$$ where $$I(0)$$ is finite but $$I(\infty)=\lim_{x\to\infty}\frac{e^x}{1+t^2}\bigg[\sin tx +\cos tx-i(t\sin tx-\cos tx)\bigg]$$ which diverges since both the real and imaginary parts oscillate between increasingly large numbers.