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How to calculate the integral of exp(-ikt) from -infinity to zero

  1. Sep 9, 2013 #1
    1. The problem statement, all variables and given/known data

    Hi everyone. I am trying to calculate $$f(k) = \int_{-\infty}^0 e^{-i k t} dt.$$

    2. Relevant equations

    Of course we know that $$\int_{-\infty}^{\infty} e^{-i k t} dt = 2 \pi \delta(k)$$ and in fact (I think) I know that the answer to my question must be $$f(k) = \frac{i}{k} + \pi \delta(k).$$ But I am puzzled as to how exactly one can show that this is the answer.

    3. The attempt at a solution

    I assume we need to make a substitution $$t \rightarrow t + i\epsilon,$$ use complex analysis, and eventually let $$\epsilon \rightarrow 0.$$ But how exactly do we do that? What theorem in complex analysis do we use and what would the result be before epsilon is set to zero?
     
  2. jcsd
  3. Sep 9, 2013 #2
    This is a nice but sort of tough question. Here are some hints for you.

    First, doing t→t+iε doesn't seem to lead anywhere very helpful. However, you should try k→k+iε. To make it look nicer I changed variables to x=-t. You can factor the exponential in the integrand into two terms, one like eikx and another term like e-εx. Now try doing integration by parts. As with other trigonometric integrals, when you change it into the form uv-∫vdu, it turns out that second integral is related to the integral you want. You need to do a little algebra and be careful about taking limits, and you should get an answer similar to (but not exactly) what you guessed.
     
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