Cauchy–Riemann equations

Characteristic property of holomorphic functions

In mathematics, the Cauchy–Riemann equations are two partial differential equations that characterize differentiability of complex functions. The equations are and where u(x, y) and v(x, y) are real bivariate differentiable functions. Typically, u and v are the real and imaginary parts, respectively, of a complex-valued function f ( z) = f ( x + i y) = u ( x , y) + i v ( x , y) {\displaystyle f(z)=f(x+iy)=u(x,y)+iv(x,y)} of a complex variable z = x + iy.

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