How meaningful is it to try to integrate around the branch-cut of a function?
For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ia$ and moving up and down the $y-$axis respectively. Now I am trying to integrate around a small circle around such a branch-point by avoiding crossing the branch-cut.
- Like can I say,
$\lim_{\epsilon \rightarrow 0} [ [\int_{\phi = \pi/2} ^{\phi = -\pi } + \int_{\phi = \pi} ^{\phi = \pi/2} ] z\text{ }\tanh(\pi z)\log(z^2+a^2) dz ] = 0 $
(where $z = ia + \epsilon e^{i\phi}$)
?
- Or at least can one argue (by some symmetry or parity argument?) the sum of such integrals around the two branch-points $\pm ia$ to be zero where one goes clockwise around both? (..and the angle integrals around each is split and ordered so as to avoid the respective branch-cuts..)
Like can I say,
$\lim_{\epsilon \rightarrow 0} \left [ [\int_{\phi_1 = \pi/2} ^{\phi_1 = -\pi } + \int_{\phi_1 = \pi} ^{\phi_1 = \pi/2} ] z_1\text{ }\tanh(\pi z_1)\log(z_1^2+a^2) dz_1 + [\int_{\phi_2 = \pi} ^{\phi_2 = -\pi/2 } + \int_{\phi_2 = -\pi/2} ^{\phi_2 = -\pi} ] z_2\text{ }\tanh(\pi z_2)\log(z_2^2+a^2) dz_2 \right ] = 0 $
(where $z_1 = ia + \epsilon e^{i\phi_1}$ and $z_2 = -ia + \epsilon e^{i\phi_2}$ )
?
I understand that this query might not be really research level but its something about which asking around I am not getting a clear answer. I had this question also lying around unanswered for days on Mathematics Stackexchange.
