Cauchy–Riemann equations

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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. If f is complex-differentiable at a complex point z = x + iy, then the partial derivatives of u and v exist and satisfy the Cauchy–Riemann equations at that point. Conversely, if the functions u and v are (real) differentiable at z and satisfy the Cauchy-Riemann equations there, then...

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