Jackson integral
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. For methods of numerical evaluation, see [1] and (Exton 1983).
Definition
Let f(x) be a function of a real variable x. For a a real variable, the Jackson integral of f is defined by the following series expansion:
- [math]\displaystyle{ \int_0^a f(x)\,{\rm d}_q x = (1-q)\,a\sum_{k=0}^{\infty}q^k f(q^k a). }[/math]
Consistent with this is the definition for [math]\displaystyle{ a \to \infty }[/math]
[math]\displaystyle{ \int_0^\infty f(x)\,{\rm d}_q x = (1-q)\sum_{k=-\infty}^{\infty}q^k f(q^k ). }[/math]
More generally, if g(x) is another function and Dqg denotes its q-derivative, we can formally write
- [math]\displaystyle{ \int f(x)\,D_q g\,{\rm 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)\tfrac{g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^k x}, }[/math] or
- [math]\displaystyle{ \int f(x)\,{\rm d}_q g(x) = \sum_{k=0}^{\infty} f(q^k x)\cdot(g(q^{k}x)-g(q^{k+1}x)), }[/math]
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 (see [2]).
Theorem
Suppose that [math]\displaystyle{ 0\lt q\lt 1. }[/math] If [math]\displaystyle{ |f(x)x^\alpha| }[/math] is bounded on the interval [math]\displaystyle{ [0,A) }[/math] for some [math]\displaystyle{ 0\leq\alpha\lt 1, }[/math] then the Jackson integral converges to a function [math]\displaystyle{ F(x) }[/math] on [math]\displaystyle{ [0,A) }[/math] which is a q-antiderivative of [math]\displaystyle{ f(x). }[/math] Moreover, [math]\displaystyle{ F(x) }[/math] is continuous at [math]\displaystyle{ x=0 }[/math] with [math]\displaystyle{ F(0)=0 }[/math] and is a unique antiderivative of [math]\displaystyle{ f(x) }[/math] in this class of functions.[3]
Notes
- ↑ Exton, H (1979). "Basic Fourier series". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 369 (1736): 115–136. doi:10.1098/rspa.1979.0155. Bibcode: 1979RSPSA.369..115E.
- ↑ Kempf, A; Majid, Shahn (1994). "Algebraic q-Integration and Fourier Theory on Quantum and Braided Spaces". Journal of Mathematical Physics 35 (12): 6802–6837. doi:10.1063/1.530644. Bibcode: 1994JMP....35.6802K.
- ↑ Kac-Cheung, Theorem 19.1.
References
- 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.
- Exton, Harold (1983). Q-hypergeometric functions and applications. Chichester [West Sussex]: E. Horwood. ISBN 978-0470274538.
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