Cauchy–Riemann equations

The equations relating the first partial derivatives of a complex function’s real and imaginary parts. With continuous first partial derivatives, they characterize complex differentiability.

Complex differentiability: A complex function is complex differentiable at a point when its difference quotient has a limit independent of direction. The equations characterize this property when the function has continuous first partial derivatives.
Complex number: A complex number has the form x + iy, where x and y are real and i² = −1. The equations arise by expressing both input and output complex numbers in real coordinates.
Augustin-Louis Cauchy: Augustin-Louis Cauchy was a French mathematician whose work established major foundations of analysis. His work on complex functions helped establish conditions associated with complex differentiability.
Real differentiability: A function between real vector spaces is real differentiable when it has a linear first-order approximation at a point. Real differentiability alone does not require the special Jacobian structure imposed by these equations.
Holomorphic function: A complex function is holomorphic on an open set when it is complex differentiable at every point in that set. The equations test holomorphicity when their regularity conditions hold throughout an open set.
Complex plane: The complex plane represents complex numbers as points with real and imaginary coordinates. Its two coordinate directions produce the paired partial derivatives in the equations.
Bernhard Riemann: Bernhard Riemann was a German mathematician whose work transformed analysis, geometry, and number theory. His complex-function theory made the equations central to the study of analytic functions.
Wirtinger calculus: Wirtinger calculus treats a complex variable and its conjugate as formal variables for differentiating functions on the plane. It reformulates the equations as a concise condition on one of its formal derivatives.
Wirtinger derivatives: Wirtinger derivatives express differentiation with respect to a complex variable and its conjugate using real partial derivatives. The equations are equivalent to the vanishing of the derivative with respect to the conjugate variable.
Partial derivative: A partial derivative measures how a function changes with respect to one variable while holding others fixed. Each equation matches derivatives taken along the real and imaginary coordinate directions.

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