I was experimenting with some complex analysis and i have some problems with my considerations.
Consider $f$ holomorphic ,with $f : \Omega_1\to \Omega_2 ,$ where both $\Omega_1,\Omega_2$ are open sets that contain the unit disc $\mathbb{D} $ ,then using the fact that $z=\frac{1}{\bar{z}}$ and Green's theorem for every ,$ a\ne z ,$ with $|a|<1$ , I find that: $$\int_{\partial \mathbb{D}}\left(-\frac{1}{z}+\log\left(1-\frac{a}{z}\right) \left(\bar{a}+\frac{1}{a}\right)\right)f(z)dz=2i\int\int_{\mathbb{D}}\frac{\bar{z}-\bar{a}}{1-a\bar{z}}f(z)dxdy.$$ I have done all the calculations (not so many but that is not my question) and I find the above result which also can be written (using Cauchy's integral formula) as: $$2i\int\int_{\mathbb{D}}\frac{\bar{z}-\bar{a}}{1-a\bar{z}}f(z)dxdy=\left(\bar{a}+\frac{1}{a}\right)\int_{\partial \mathbb{D}}\left(\log\left(1-\frac{a}{z}\right) \right)f(z)dz-\frac{f(0)}{a}.$$
MY QUESTION: Do i have the right to use Green's theorem here? Is the logarithm here well defined ? Can someone verify, or help me with this one?
Yes, the logarithm is well defined.
Since $|\frac{a}{z}|< 1$ for $a$ with $|a|<1$ and $ z\in \partial \mathbb{D}$, we have $$ \operatorname{Re}\, \left(1-\frac{a}{z}\right)>0.$$ Thus $\log \left(1-\frac{a}{z}\right)$ is well defined for $ z\in \partial \mathbb{D}$ (NOT for $z\in\mathbb{D}$) and the expression$$ \int_{\partial \mathbb{D}}\left(-\frac{1}{z}+\log\left(1-\frac{a}{z}\right) \left(\bar{a}+\frac{1}{a}\right)\right)f(z)dz $$ is defined meaningfully.
Also you have the right to use Green's theorem.
Since $z\bar{z}=1$ for $z\in\partial \mathbb{D}$ we can rewrite $$ \int_{\partial \mathbb{D}}\left(-\frac{1}{z}+\log\left(1-\frac{a}{z}\right) \left(\bar{a}+\frac{1}{a}\right)\right)f(z)dz $$ as $$\int_{\partial \mathbb{D}}\left(-\bar{z}+\log\left(1-a\bar{z}\right) \left(\bar{a}+\frac{1}{a}\right)\right)f(z)dz. $$ Since $\log (1-a\bar{z})$ is well defined for all $z\in\mathbb{D}$, we can use Green's theorem to get \begin{align} \int_{\partial \mathbb{D}}\left(-\bar{z}+\log\left(1-a\bar{z}\right) \left(\bar{a}+\frac{1}{a}\right)\right)f(z)dz&=2i\iint_\mathbb{D}\frac{\partial}{\partial \bar{z}}\left\{\left(-\bar{z}+\log\left(1-a\bar{z}\right) \left(\bar{a}+\frac{1}{a}\right)\right)f(z)\right\}dxdy\\ &=2i\iint_\mathbb{D}\left(-1+\frac{-a}{1-a\bar{z}}\cdot\left(\bar{a}+\frac{1}{a}\right)\right)f(z)\,dxdy. \end{align}