Jackson integral

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In q-analog theory, the Jackson integral series in the theory of special functions that expresses the operation inverse to q-differentiation.

The Jackson integral was introduced by Frank Hilton Jackson.

Definition[edit]

Let f(x) be a function of a real variable x. The Jackson integral of f is defined by the following series expansion:

 \int f(x) d_q x = (1-q)x\sum_{k=0}^{\infty}q^k f(q^k x).

More generally, if g(x) is another function and Dqg denotes its q-derivative, we can formally write

 \int f(x) D_q g d_q x = (1-q)x\sum_{k=0}^{\infty}q^k f(q^k x) D_q g(q^k x) = (1-q)x\sum_{k=0}^{\infty}q^k f(q^k x)\frac{g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^k x}, or
 \int f(x) d_q g(x) = \sum_{k=0}^{\infty} f(q^k x)(g(q^{k}x)-g(q^{k+1}x)),

giving a q-analogue of the Riemann–Stieltjes integral.

Jackson integral as q-antiderivative[edit]

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.

Theorem[edit]

Suppose that 0<q<1. If |f(x)x^\alpha| is bounded on the interval [0,A) for some 0\leq\alpha<1, then the Jackson integral converges to a function F(x) on [0,A) which is a q-antiderivative of f(x). Moreover, F(x) is continuous at x=0 with F(0)=0 and is a unique antiderivative of f(x) in this class of functions.[1]

Notes[edit]

  1. ^ Kac-Cheung, Theorem 19.1.

References[edit]

  • 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.