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MATHEMATICS Complex Analysis Cauchy–Riemann Equations

Cauchy–Riemann Equations

Cauchy–Riemann equations reveal the hidden harmony between real and imaginary parts of complex functions

3 Nov 2025

Why do the Cauchy–Riemann equations guarantee that a complex function is analytic, not just differentiable?

Because the equations imply the function is continuous everywhere on the complex plane. Because the equations only require the function to have real derivatives, which is sufficient for analyticity. Because satisfying the Cauchy–Riemann equations ensures the function can be locally represented by a convergent power series.