Indefinite sum

In mathematics the indefinite sum operator (also known as the antidifference operator), denoted by ∑ x {\displaystyle \sum _{x}} or Δ − 1 {\displaystyle \Delta ^{-1}} , is the linear operator, inverse of the forward difference operator Δ {\displaystyle \Delta } . It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus Δ ∑ x f ( x ) = f ( x ) . {\displaystyle \Delta \sum _{x}f(x)=f(x)\,.} More explicitly, if ∑ x f ( x ) = F ( x ) {\displaystyle \sum _{x}f(x)=F(x)} , then F ( x + 1 ) − F ( x ) = f ( x ) . {\displaystyle F(x+1)-F(x)=f(x)\,.} If F(x) is a solution of this functional equation for a given f(x), then so is F(x)+C(x) for any periodic function C(x) with period 1. Therefore each indefinite sum actually represents a family of functions. However the solution equal to its Newton series expansion is unique up to an additive constant C. This unique solution can be represented by formal power series form of the antidifference operator: Δ − 1 = 1 e D − 1 {\displaystyle \Delta ^{-1}={\frac {1}{e^{D}-1}}}

Words

This table shows the example usage of word lists for keywords extraction from the text above.

WordWord FrequencyNumber of ArticlesRelevance
x24298920.689
f17448430.45
antidifference320.245
operator7108130.24
indefinite59940.237

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