Is $z\ln z$ an analytic complex function in $z=0$?

Question

I want to calculate the following Integral $$\oint _{|z-i|=3}\frac{z\ln z}{(z-2i)}dz.$$ for this, The Cauchy's Integral Formula is good method. But this can be applied when the $z\ln z$ is analytic inside of the given region. Can I use this theorem? i.e. Is $z\ln z$ an analytic complex function in $z=0$?

Answer 1

The result depends on which branch of the logarithm you choose. With the principal branch, the integral taken in the ccw direction is equal to $$2 \pi i \operatorname*{Res}_{z = 2 i} f(z) - \int_{-2 \sqrt 2 + i 0}^{i 0} f(z) dz + \int_{-2 \sqrt 2 - i 0}^{-i 0} f(z) dz = \\ 2 \pi i \left( 2 i \ln 2 i - \int_{-2 \sqrt 2}^0 \frac z {z - 2 i} dz \right).$$