Cauchy-Riemann equations

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In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations that provides a necessary and sufficient condition for a differentiable function to be holomorphic in an open set. This system of equations first appeared in the work of Jean le Rond d'Alembert (d'Alembert 1752). Later, Leonhard Euler connected this system to the analytic functions (Euler 1797). Cauchy (1814) then used these equations to construct his theory of functions. Riemann's dissertation (Riemann 1851) on the theory of functions appeared in 1851.

The Cauchy–Riemann equations on a pair of real-valued functions u(x,y) and v(x,y) are the two equations:

and

Typically u and v are taken to be the real and imaginary parts respectively of a complex-valued function f(x + iy) = u(x,y) + iv(x,y). Suppose that u and v are continuously differentiable on an open subset of C. Then f = u+iv is holomorphic if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann equations (1a) and (1b).

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