How do we evaluate the integral $$\int_0^1\frac{\ln z}{z-1}\,\mathrm dz\,?$$ Here contour of integration consists of \begin{align*} \gamma_R(t)&=1+R\mathrm e^{\mathrm it}\quad&\text{for}\quad&t\in[0,\pi],\\ l_1(t)&=t&\text{for}\quad&t\in[1-R,1-\epsilon],\\ \gamma_\epsilon(t)&=1+\epsilon\mathrm e^{-\mathrm it}&\text{for}\quad&t\in[-\pi,0]\\ \text{and}\quad l_2(t)&=t&\text{for}\quad&t\in[1+\epsilon,1+R]. \end{align*} Am I right so far? How should I approach the rest of the calculation Any help is welcome, thanks in advance.
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2Substitute $z=e^t$ and transform to a zeta function integral. – Integrand Nov 03 '21 at 17:40
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This is a very well known integral with multiple solution methods, see this question for instance. – K.defaoite Nov 03 '21 at 17:49