Let $D=\mathbb{C}\sim(-\infty,0]$. Define $f:D\rightarrow \mathbb{C}$ by $f(z)=(z-1)/Log(z)$ if $z\neq 1$, while $f(1)=1$. verify that $f$ is an analytic function.
I do not know very well how to prove this, I tried in the following way: Proving that it is real differentiable and that the Cauchy Riemann equations are fulfilled in the whole plane but I consider this difficult because I would have to express $f$ as $f=u+iv$. I know that $g(z)=Log(z)$ is analytic in $\mathbb{C}\sim(-\infty,0]$ and that $h(z)=z-1$ is entire, how can I use this to prove that $f$ is analytic? Thank you very much.