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Cauchy's functional equation is the functional equation: (+) = + (). A function that solves this equation is called an additive function.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely : for any rational constant .
Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function is a function that satisfies the functional equation f ( x + 1 ) = x f ( x ) {\displaystyle f(x+1)=xf(x)} and the initial value f ( 1 ) = 1. {\displaystyle f ...
In the special case when Ulam's problem accepts a solution for Cauchy's functional equation f(x + y) = f(x) + f(y), the equation E is said to satisfy the Cauchy–Rassias stability. The name is referred to Augustin-Louis Cauchy and Themistocles M. Rassias .
That is, the Cauchy–Riemann equations are the conditions for a function to be conformal. Moreover, because the composition of a conformal transformation with another conformal transformation is also conformal, the composition of a solution of the Cauchy–Riemann equations with a conformal map must itself solve the Cauchy–Riemann equations.
The stability problem of functional equations originated from a question of Stanisław Ulam, posed in 1940, concerning the stability of group homomorphisms.In the next year, Donald H. Hyers [1] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces in the case of additive mappings, that was the first significant breakthrough and a step toward more solutions ...
Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide. ... Cauchy's functional equation; F. Functional equation (L ...
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem. [7] Naturally also same function of one variable that depends on some complex parameter is a candidate.