In analogy to the indefinite integral being the inverse operator to the derivative, we now define a similar concept for the difference operator.

Definition: Indefinite Sum, Antidifference

For a given complex-valued function $F:\mathbb C\to\mathbb C$, let the function $f:\mathbb C\to\mathbb C$ be defined by the difference operator $$f(x):=\Delta F(x).$$

The indefinite sum is the operation $$\sum f(x):=\Delta^{-1}f(x)=F(x).$$ Because of this property, the function $F$ is also called the antidifference of $f.$

  1. Proposition: Antidifferences are Unique Up to a Periodic Constant

Chapters: 1
Proofs: 2 3 4
Propositions: 5 6
Theorems: 7

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  1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
  2. Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
  3. Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960