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:
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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?
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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.
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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.
Best Answer
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.