Cauchy-Riemann equations
In complex analysis, the Cauchy-Riemann differential equations are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic.
Let f = u + iv be a function from an open subset of the complex numbers C to C, and regard u and v as real-valued functions defined on an open subset of R2. Then f is holomorphic if and only if u and v are differentiable and their partial derivatives satisfy the Cauchy-Riemann equations, which are:
and
 .
It follows from the equations that u and v must be harmonic functions. The equations can therefore be seen as the conditions on a given pair of harmonic functions to come as real and imaginary parts of a complex-analytic function.
Referenced By
Atiyah-Singer index theorem | Atiyah-Singer theorem | Augustin-Louis Cauchy | Augustin Cauchy | Augustin Louis Cauchy | Cauchy | Holomorphic function | Hydrodynamics | List of complex analysis topics | List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | Potential flow | Potential flow in 2d) | Potential flow in two dimensions | Several complex variables
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