# Nested Cauchy type integral

I have a function which is in integral form:

$$f_+(z)=\exp\bigg(\frac{1}{2\pi i}\oint_{C}\frac{f(\alpha)}{(z-\alpha)}\,d\alpha\bigg),$$

where $$C$$ is a unit circle inside an annulus in $$z$$ complex plane. I want to take the inverse Fourier transform (it is the inverse of a discrete Fourier transform) of this function, that is,

$$F(x)=\frac{1}{2\pi i}\oint_Cf_{+}(z)z^{x-1}\,dz,$$

where again $$C$$ is a unit circle. The function $$f(z)$$ has a complicated expression, so I am avoiding writing it here. How can I do such integration numerically?

Thank you.