The Jackson integral was introduced by Frank Hilton Jackson.
Let f(x) be a function of a real variable x. The Jackson integral of f is defined by the following series expansion:
More generally, if g(x) is another function and Dqg denotes its q-derivative, we can formally write
giving a q-analogue of the Riemann–Stieltjes integral.
Jackson integral as q-antiderivative
Just as the ordinary antiderivative of a continuous function can be represented by its Riemann integral, it is possible to show that the Jackson integral gives a unique q-antiderivative within a certain class of functions.
Suppose that If is bounded on the interval for some then the Jackson integral converges to a function on which is a q-antiderivative of Moreover, is continuous at with and is a unique antiderivative of in this class of functions.
- Kac-Cheung, Theorem 19.1.
- Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
- Jackson F H (1904), "A generalization of the functions Γ(n) and xn", Proc. R. Soc. 74 64–72.
- Jackson F H (1910), "On q-definite integrals", Q. J. Pure Appl. Math. 41 193–203.