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I am working my way through some complex analysis notes(from a physics course). I have just covered Cauchy's theorem which basically states that the integral over a closed contour of an analytic function is zero. this is then used to show that contours of analytic functions can be deformed and the integrals still give the same answer.

My problem begins when Cauchy's integral formula is introduced and it is used to compute an integral of 1/z around a unit circle. The notes then state that an arbitrary curve can be deformed to a circle using Cauchy's theorem but this theorem is only valid for analytic functions and 1/z is not analytic at the origin. So how can arbitrary curves be deformed ?

Thanks