Let $C$ be the contour of the unit square with lower left corner at origin. We define a function $g(z)=\int_{z+C} f(w)dw$ for a given (not necessarily holomorphic) function $f:\mathbb{C}\to\mathbb{C}$.
My question: If we know $g$ is injective, is there any way to compute the inverse of $g$? Should we expect an analytic expression, or do we need some numerical methods?
I am particularly interested in the case when $f(w)=\frac{\bar w}{w}$ (which comes from a physics experiment). In this case $g(a+bi)=const (A+Bi)$ where
$A=\int_b^{b+1}\int_a^{a+1}\frac{x}{x^2+y^2}dxdy$ and $B=\int_b^{b+1}\int_a^{a+1}\frac{y}{x^2+y^2}dxdy$
can be computed easily. We cannot see the expression of the inverse of $g$ from that and would like some insights and suggestions, analytically or numerically.
Thank you.
