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Let's say we have an integral of the form $I = \int_X f(z) dz$ where $X$ is some subset of the complex numbers, is there a methods to numerically evaluate this integral?

For simplicity we can of assume that $X$ is a polygon in the complex plane. (If there are any methods for other shapes of the integration domain please let me know too, that was just intended as an example.)

I'm especially interested in cases where $X$ is unboundend, i.e. $X = \mathbb C$ or $X = \{ z \in \mathbb C \mid Im(z)>0 \}$

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    Can you write a Riemann sum or similar for your integral, assuming that $f$ is smooth enough? If $F$ is an anti-derivative of $F$, can the integral be expressed in terms of $F$? You are integrating over a 2 dimensional domain with a 1D curve integral, what do expect to get? – 2017-01-16
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    I'm integrating over a 2 dimensional domain, it is not a path integral. $f$ should just be any sufficiently smooth function $f: \mathbb C \to \mathbb C$ – 2017-01-16
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    But $dz=dx+i·dy$ is a one-dimensional curve element, not a two-dimensional area element like $dx\,dy$. Already your notation does not make sense, thus the question for an integral definition. – 2017-01-16
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    The complex numbers can be considered as one dimensional, and I have seen this notation being used frequently. For example [here](https://en.wikipedia.org/wiki/Schwarz%E2%80%93Christoffel_mapping). Some people even entirely omit the $dz$. What notation would you suggest using instead? – 2017-01-16
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    In that case, the integral is along the boundary curves of the polygon, which is completely in the sense of the notation. – 2017-01-16

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