How is the contour integral with absolute value performed?
Here $$\mathfrak{I}=\int_{C_1} \frac{a|z|}{z-\gamma}\mathrm{d}z$$ where the contour $C_1$ is parallel to the real line but passes above the point $z=\gamma$ and $a\in \mathbb{C}$
Is taking $z=R e^{i\theta}$ and hence $\mathrm{d}z=R i e^{i\theta}$ and taking the limit from $0$ to $\pi$ a correct approach?
That gives $$\mathfrak{I}=\int_0^\pi \frac{aR}{Re^{i\theta}-\gamma}R i e^{i\theta}\mathrm{d}\theta=a R (\log (-\gamma -R)-\log (R-\gamma ))$$
but it does not seem to converge in the limit $R\to \infty$.
It's a little unclear. Given the contour $C_1=x+i k$ for constant k, then looks like your integral is: $$\int_{x_1}^{x_2} \frac{a\sqrt{x^2+k^2}}{(x+ik)-\gamma}dx$$ $$=\biggr\{a \left(-\sqrt{\gamma } \sqrt{-\gamma +2 i k} \tan ^{-1}\left(\frac{k^2-i k x+\gamma x}{\sqrt{\gamma } \sqrt{-\gamma +2 i k} \sqrt{k^2+x^2}}\right)+(\gamma -i k) \tanh ^{-1}\left(\frac{x}{\sqrt{k^2+x^2}}\right)+\sqrt{k^2+x^2}\right)\biggr\}\biggr|_{x1}^{x_2}$$
If it were mine, I would evaluate it numerically say from 1 to 10, then compare the results with the expression above. However the antiderivative has multi-valued functions which will cause problems with a definite integral