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I am aware of the Sokhotski–Plemelj theorem (I have also heard people referring to it as the "Dirac identity") which states that in the limit $\eta\rightarrow 0^+$

$$\frac{1}{x\pm i\eta}=\mathcal P\frac{1}{x}\mp i\pi\delta(x) \, .$$

Now, I am reading the book "Solid State Physics" by G. Grosso and G. Pastori Parravicini which states on page 430 that using the above formula it can "easily be proved" that

$$\frac{\hbar\omega}{E_j-E_i-\hbar\omega-i\eta}= \frac{E_j-E_i}{E_j-E_i-\hbar\omega-i\eta}-1\, .$$

However, I fail to see how the latter formula follows from the former. Is there a trick that I am missing here?

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Noting the simple relation $$ \frac{\Delta E_{i,j}}{\Delta E_{i,j}-\hbar\omega-i\eta}-\frac{\Delta E_{i,j}-\hbar\omega-i\eta}{\Delta E_{i,j}-\hbar\omega-i\eta}=\frac{\hbar\omega+i\eta}{\Delta E_{i,j}-\hbar\omega-i\eta} $$ and by Sokhotski-Plemelj theorem $$ \lim_{\eta\rightarrow 0^+} \frac{i\eta}{\Delta E_{i,j}-\hbar\omega-i\eta}=0 $$ because, if the limit $$ \lim_{\eta\rightarrow0^+}\int_{a}^{b}\frac{f(x)}{x\pm i\eta}dx $$ exist, that's what the theorem says, then $$ \lim_{\eta\rightarrow0^+}\int_{a}^{b}\frac{i\eta f(x)}{x\pm i\eta}dx= \left(\lim_{\eta\rightarrow0^+} i\eta\right)\left( \lim_{\eta\rightarrow0^+}\int_{a}^{b}\frac{f(x)}{x\pm i\eta}dx\right)=0 $$

for any target function $f(x)$.

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  • $\begingroup$ Why does the last part follow from the Sokhotski-Plemelj theorem? Also, there's a sign mistake in the second equation. Also, I don't think you need the first equation at all, as the second one is trivially true. $\endgroup$ – DanielSank Feb 2 '16 at 3:34
  • $\begingroup$ @DanielSank I updated my answer correcting the sign mistake and explaining in detail the application of Sokhotski-Plemelj theorem. $\endgroup$ – Nogueira Feb 2 '16 at 4:34
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I don't think we need Sokhotski-Plemelj for this. Think of $E_j - E_i$ as a fixed value $E$. Then the formula is re-written as

$$\frac{\hbar \omega}{E - \hbar \omega - i \eta}\, .$$

Now let $x \equiv \hbar \omega$ and you get

$$\frac{x}{E -x - i \eta} \, .$$

This integral is dominated by the part where $x \approx E$ so let's try shifting the variables $y \equiv E - x$,

$$\frac{E - y}{y - i \eta}$$

and then expand the numerator and put the original variables back in:

$$\frac{E_j - E_i}{E_j - E_i - \hbar \omega - i \eta} - \frac{E_j - E_i - \hbar \omega}{E_j - E_i - \hbar \omega - i \eta} \, .$$

The first term already matches the first term in the target expression, so we only need to worry about the second term. The second term is 1 because, well, in the limit $\eta \rightarrow 0$ it's identically 1.

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