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- Cauchy–Riemann_equations abstract "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, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is holomorphic. 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 of two real variables u(x,y) and v(x,y) are the two equations:(1a) (1b) Typically u and v are taken to be the real and imaginary parts respectively of a complex-valued function of a single complex variable z = x + iy, f(x + iy) = u(x,y) + iv(x,y). Suppose that u and v are real-differentiable at a point in an open subset of C ( C is the set of complex numbers), which can be considered as functions from R2 to R. This implies that the partial derivatives of u and v exist (although they need not be continuous) and we can approximate small variations of f linearly. Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann equations (1a) and (1b) at that point. The sole existence of partial derivatives satisfying the Cauchy–Riemann equations is not enough to ensure complex differentiability at that point. It is necessary that u and v be real differentiable, which is a stronger condition than the existence of the partial derivatives, but it is not necessary that these partial derivatives be continuous.Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of C (this is called a domain in C). Consequently, we can assert that a complex function f, whose real and imaginary parts u and v are real-differentiable functions, is holomorphic if and only if, equations (1a) and (1b) are satisfied throughout the domain we are dealing with.The reason why Euler and some other authors relate the Cauchy–Riemann equations with analyticity is that a major theorem in complex analysis says that holomorphic functions are analytic and vice versa. This means that, in complex analysis, a function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.".
- Cauchy–Riemann_equations wikiPageID "7583".
- Cauchy–Riemann_equations wikiPageRevisionID "593903265".
- Cauchy–Riemann_equations first "E.D.".
- Cauchy–Riemann_equations id "c/c020970".
- Cauchy–Riemann_equations last "Solomentsev".
- Cauchy–Riemann_equations title "Cauchy–Riemann Equations".
- Cauchy–Riemann_equations title "Cauchy–Riemann conditions".
- Cauchy–Riemann_equations urlname "Cauchy-RiemannEquations".
- Cauchy–Riemann_equations year "2001".
- Cauchy–Riemann_equations subject Category:Complex_analysis.
- Cauchy–Riemann_equations subject Category:Equations.
- Cauchy–Riemann_equations subject Category:Harmonic_functions.
- Cauchy–Riemann_equations subject Category:Partial_differential_equations.
- Cauchy–Riemann_equations comment "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, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is holomorphic. This system of equations first appeared in the work of Jean le Rond d'Alembert (d'Alembert 1752).".
- Cauchy–Riemann_equations label "Cauchy-Riemann-vergelijkingen".
- Cauchy–Riemann_equations label "Cauchy-Riemannsche partielle Differentialgleichungen".
- Cauchy–Riemann_equations label "Cauchy–Riemann equations".
- Cauchy–Riemann_equations label "Ecuaciones de Cauchy-Riemann".
- Cauchy–Riemann_equations label "Equazioni di Cauchy-Riemann".
- Cauchy–Riemann_equations label "Równania Cauchy’ego-Riemanna".
- Cauchy–Riemann_equations label "Équations de Cauchy-Riemann".
- Cauchy–Riemann_equations label "Условия Коши — Римана".
- Cauchy–Riemann_equations label "معادلات كوشي-ريمان".
- Cauchy–Riemann_equations label "柯西－黎曼方程".
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- Cauchy–Riemann_equations sameAs 코시-리만_방정식.
- Cauchy–Riemann_equations sameAs Cauchy-Riemann-vergelijkingen.
- Cauchy–Riemann_equations sameAs Równania_Cauchy’ego-Riemanna.
- Cauchy–Riemann_equations sameAs Q622741.
- Cauchy–Riemann_equations sameAs Q622741.
- Cauchy–Riemann_equations wasDerivedFrom Cauchy–Riemann_equations?oldid=593903265.