termwise differentiation


If in the open interval I, all the of the series

f1(x)+f2(x)+ (1)

have continuousMathworldPlanetmathPlanetmath derivativesPlanetmathPlanetmath, the series convergesPlanetmathPlanetmath having sum S(x) and the differentiated series  f1(x)+f2(x)+converges uniformly (http://planetmath.org/SumFunctionOfSeries) on the interval I, then the series (1) can be differentiated termwise, i.e. in every point of I the sum function S(x) is differentiableMathworldPlanetmathPlanetmath and


The situation implies also that the series (1) converges uniformly on I.

Title termwise differentiation
Canonical name TermwiseDifferentiation
Date of creation 2013-03-22 14:38:38
Last modified on 2013-03-22 14:38:38
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 9
Author Mathprof (13753)
Entry type Theorem
Classification msc 26A15
Classification msc 40A30
Synonym differentiating a series
Related topic PowerSeries
Related topic IntegrationOfLaplaceTransformWithRespectToParameter
Related topic IntegralOfLimitFunction