Why does complex differentiability impose stronger conditions than real differentiability?
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.
Complex differentiability imposes stronger conditions than real differentiability because of the geometric nature of the complex plane. While a real variable can only approach a point from two directions (left or right) along a line, a complex variable can approach a point from an infinite number of directions in the plane.1
For a complex function to be differentiable, the limit of the difference quotient must be the same regardless of the path taken as the increment tends to zero. This requirement forces the real and imaginary parts of the function to satisfy specific constraints, known as the Cauchy-Riemann equations.1
Because of these rigid constraints, complex differentiability is a much more powerful property than in real analysis. If a complex function is differentiable in a region, it is automatically infinitely differentiable and can be represented by a power series, making it analytic. In contrast, real differentiable functions are far more flexible and do not necessarily possess these properties. Consequently, the values of a complex differentiable function in any small region determine the entire function, a level of rigidity absent in real analysis.21