The following theorem can be called the fundamental theorem of difference calculus, as it provides a discrete version of the fundamental theorem of calculus.

# Theorem: Fundamental Theorem of the Difference Calculus

Let $F:D\to\mathbb C$ be an antidifference of a complex-valued function $f:C\to\mathbb C$, $D\subseteq\mathbb C.$ Then, for all $a,b\in C$ and all natural numbers $n\in\mathbb N$ the following formula holds:

$$\sum_{x=a}^{a+n-1}f(x)=F(a+n)-F(a).$$

### Different Notation

$$\sum_{x=a}^{a+n-1}f(x)=F(x)\;\Rule{1px}{4ex}{2ex}^{a+n}_{a}=\Delta^{-1}f(x)\;\Rule{1px}{4ex}{2ex}^{a+n}_{a}.$$

Proofs: 1

Chapters: 1 2
Problems: 3 4 5 6 7
Solutions: 8

Github: ### References

#### Bibliography

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