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Concept
Cauchy–Riemann equations
Summary
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable.
These equations are
and
where u(x, y) and v(x, y) are real differentiable bivariate functions.
Typically, u and v are respectively the real and imaginary parts of a complex-valued function f(x + iy) = f(x, y) = u(x, y) + iv(x, y) of a single complex variable z = x + iy where x and y are real variables; u and v are real differentiable functions of the real variables.
Then f is complex differentiable at a complex point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann equations at that point.
A holomorphic function is a complex function that is differentiable at every point of some open subset of the complex plane C. It has been proved that holomorphic functions are analytic and analytic complex functions are complex-differentiable. In particular, holomorphic functions are infinitely complex-differentiable.
This equivalence between differentiability and analyticity is the starting point of all complex analysis.
The Cauchy–Riemann equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler connected this system to the analytic functions. Cauchy then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.
Suppose that . The complex-valued function is differentiable at any point z in the complex plane.
The real part and the imaginary part are
and their partial derivatives are
We see that indeed the Cauchy–Riemann equations are satisfied, and .
The Cauchy-Riemann equations are one way of looking at the condition for a function to be differentiable in the sense of complex analysis: in other words, they encapsulate the notion of function of a complex variable by means of conventional differential calculus.
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