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Here $z=x+iy$ is a complex variable, as is $w={\mbox{Re}}\;w+i{\mbox{Im}}\;w$ and $f(z)={\mbox{Re}}\;f+i{\mbox{Im}}\;f.$ the function is a complex-valued function of $z$. The fuction $f$ is analytic meaning that the derivative of $f(z)$ with respect to $z$ exists (and similarly the derivative with respect to $w$ of $f(w)$ exists) for some region bounded by the closed curve $C$ called the contour of integration.

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