Practice problem

Complex Analysis and Riemann Surfaces

The proof of Hartogs's extension theorem is based on the following idea.

If $f(z_1,z_2)$ is a holomorphic function $\mathbb{C}^2\setminus(0,0)\to \mathbb{C}$ , then for small $(a_1,a_2)\in B^2$ , consider the function $g(a_1,a_2) = \oint_{\lvert z_1\rvert = \lvert z_2\rvert = 1} dz_1 dz_2\frac{f(z_1,z_2)}{(z_1-a_1)(z_2-a_2)}.$

What properties of $g$ are used in the proof of Hartogs's extension theorem ( $g$ is a holomorphic extension of $f$ )? Select all that apply.