Math 2.0

Difference and functional equations Definition

Indefinite sum

Builds on Forward difference operator

Definition

Let f ⁣:ℝ→ℝf \colon \R \to \R be a function, and let Δ\Delta be the forward difference operator with step 11. An indefinite sum of ff is a function F ⁣:ℝ→ℝF \colon \R \to \R with ΔF=f\Delta F = f, that is, F(x+1)−F(x)=f(x)for all x∈ℝ.F(x + 1) - F(x) = f(x) \quad \text{for all } x \in \R. Another name is antidifference. The same definition applies to functions f ⁣:ℤ→ℝf \colon \Z \to \R, with F ⁣:ℤ→ℝF \colon \Z \to \R.

Remarks

Common notations are Δ−1f\Delta^{-1} f and ∑xf(x)\sum_x f(x). Example: F(x)=2xF(x) = 2^x is an indefinite sum of f(x)=2xf(x) = 2^x, since 2x+1−2x=2x2^{x+1} - 2^x = 2^x. Non-example: F(x)=x2/2F(x) = x^2/2 is not an indefinite sum of f(x)=xf(x) = x, since F(x+1)−F(x)=x+12F(x + 1) - F(x) = x + \tfrac{1}{2}. Indefinite sums are not unique. On ℤ\Z, two indefinite sums of ff differ by a constant. On ℝ\R, two indefinite sums of ff differ by a function CC with C(x+1)=C(x)C(x + 1) = C(x) for all xx, such as sin⁡(2πx)\sin(2\pi x).

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionFinite difference, definition, no evidence levelFinite differencedefinitionForward difference operator, definition, no evidence levelForward difference op…definitionIndefinite sum, definition, no evidence levelIndefinite sumdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Existential quantifier definition
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Uniqueness quantifier definition
  12. Binary relation definition
  13. Function definition
  14. Finite difference definition
  15. Forward difference operator definition
  16. Indefinite sum definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Indefinite_sum
About this page
Tag5VK9NZM0R7
URIm2://5VK9NZM0R7
Kinddefinition
WritingW0 stub
MSC39A12, 39A70
DefinesIndefinite sum
UsesForward difference operator

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by