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

[PhilandSci]

Homework Statement

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

Homework 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.

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?

 

[Jolb]

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 e^ikx 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.