I am trying to understand a step made in a physics textbook ( Lectures on Quantum Field Theory, by Ashok Das ). But I don't even know what the formula is called, or what are the keywords. So I couldn't even google it.
On page 253, in equation (6.128 ), the author decomposes a denominator "in the standard manner" :
$$ \underset{\epsilon \rightarrow 0^+}{ lim} \frac{1}{k'_0 - k_0 - i \epsilon} = \frac{1}{k'_0 - k_0 } + i \pi \delta(k'_0 - k_0) \tag{6.128}$$
and he says that the first term on the RHS represents the principal value. Note that this denominator appears under an integral on $dk'_0$. Can anyone explain how this formula comes about? Any references are welcome.
To me, it seems that it somehow comes from complex contour integration.
( I have tried to keep it succinct and relevant to math, but I can produce more context if needed. )
It's not very difficult. We have $$ \frac{1}{t-i\epsilon} = \frac{t+i\epsilon}{t^2+\epsilon^2} = \frac{t}{t^2+\epsilon^2}+i\frac{\epsilon}{t^2+\epsilon^2} . $$ Here the first term clearly tends to $\frac{1}{t}$ as $\epsilon\to 0.$ That it is the principal value might not be as clear, but perhaps you can accept it. For the second term we integrate it against a well-behaved function $\phi$ and make a substitution $t=\epsilon s$: $$ \int \frac{\epsilon}{t^2+\epsilon^2} \phi(t) \, dt = \int \frac{\epsilon}{\epsilon^2 s^2+\epsilon^2} \phi(\epsilon s) \, \epsilon\,ds = \int \frac{1}{s^2+1} \phi(\epsilon s) \, ds \\ \to \int \frac{1}{s^2+1} \phi(0) \, ds = \pi \phi(0) = \int \pi\delta(s) \, \phi(s) \, ds . $$