What does it mean to Taylor expand $1/(\ln(r)-\ln(x)+i\pi/2)$ in power of $1/\ln(x)$?
Can some one help me understand what this text means? I could understand everything except for the last step which is to Taylor expand that part of the integrand in power of $1/\ln(x)$. I tried regularly Taylor expand it, but I encounter product rules and it does not seem to be the same as what is in the text.
You have an expression $$ \frac{1}{a-w}. $$ You can expand it as a geometric series in $\frac aw$ with the understanding that $|w|>|a|$ using the identity $$ \frac{1}{a-w}=-\frac1w\cdot\frac1{1-\frac aw}=-\frac1w⋅\sum_{k\ge 0}\left(\frac aw\right)^k. $$