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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.

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    $\begingroup$ In the case you are particularly interested, the integral can be explicitly evaluated and a formula for your $g$ obtained. Look at it and see whether it can be inverted. $\endgroup$ Nov 17, 2014 at 18:17
  • $\begingroup$ @AlexandreEremenko Thank you. We calculated that, and know it is injective. It just seems not possible to write the expression of the inverse, and we are wondering if we missed something. $\endgroup$
    – Ted Mao
    Nov 17, 2014 at 18:39

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