Cauchy–Riemann equations

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 holomo

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In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many bra

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation

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In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation