|
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 . Later, Leonhard Euler connected this system to the analytic functions . then used these equations to construct his theory of functions. Riemann's dissertation 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: = \dfrac |- |(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 , . 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 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. 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. == Interpretation and reformulation == The equations are one way of looking at the condition on 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. In the theory there are several other major ways of looking at this notion, and the translation of the condition into other language is often needed. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cauchy–Riemann equations」の詳細全文を読む スポンサード リンク
|