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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.

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