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When does Cauchy’s theorem make a complex integral independent of the chosen path?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

In complex analysis, Cauchy’s theorem establishes that the integral of an analytic function is independent of the chosen path when the integration occurs within a simply connected region.1

A simply connected region is defined as a domain that contains no "holes." When a region is simply connected, any two contours $C_{1}$ and $C_{2}$ connecting the same two points will yield the same integral value, provided the curves remain within that region.1

Conversely, if the region contains holes, the integral is generally not independent of the path. In such cases, the value of the integral depends on the topology of the curve—specifically, how many times the contour winds around the holes within the region.1