[Numerical] Computing a contour of an integral function

rsq_a

Suppose I want to compute all complex points z such that,

[tex]\int_{z^*}^z F(t) dt = 0[/tex]

Here [tex]z^*[/tex] is a given constant. In general, the points z which satisfy that relation form a continuous curve beginning from the initial point.

What is the best way to tackle this numerically? I'm sure it's a fairly standard numerical problem. Unfortunately, I'm just not sure where to look (in the literature).