Why does every differentiable complex function have derivatives of all orders?

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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, a function that is differentiable in an open region is referred to as analytic. Unlike in real analysis, where a function may be differentiable without its derivative being differentiable, a differentiable complex function is inherently smooth, meaning its derivative is also differentiable within that region.1

This property arises because such functions possess a power series expansion that converges within a disk centered at any point in the region. Because the function can be represented by this power series, it is guaranteed to have derivatives of all orders $n = 1, 2, 3, \dots$ .1