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| In [[q-analog]] theory, the '''Jackson integral''' [[series (mathematics)|series]] in the theory of [[special functions]] that expresses the operation inverse to [[q-differentiation]].
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| The Jackson integral was introduced by [[Frank Hilton Jackson]].
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| == Definition ==
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| Let ''f''(''x'') be a function of a real variable ''x''. The Jackson integral of ''f'' is defined by the following series expansion:
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| : <math> \int f(x) d_q x = (1-q)x\sum_{k=0}^{\infty}q^k f(q^k x). </math>
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| More generally, if ''g''(''x'') is another function and ''D''<sub>''q''</sub>''g'' denotes its ''q''-derivative, we can formally write
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| : <math> \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}, </math> or
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| : <math> \int f(x) d_q g(x) = \sum_{k=0}^{\infty} f(q^k x)(g(q^{k}x)-g(q^{k+1}x)), </math>
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| giving a ''q''-analogue of the [[Riemann–Stieltjes integral]].
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| == Jackson integral as q-antiderivative ==
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| 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
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| within a certain class of functions.
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| === Theorem ===
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| Suppose that <math>0<q<1.</math> If <math>|f(x)x^\alpha|</math> is bounded on the interval <math>[0,A)</math> for some <math>0\leq\alpha<1, </math> then the Jackson integral converges to a function <math>F(x)</math> on <math>[0,A)</math> which is a ''q''-antiderivative of <math>f(x).</math> Moreover, <math>F(x)</math> is continuous at <math>x=0</math> with <math>F(0)=0</math> and is a unique antiderivative of <math>f(x)</math> in this class of functions.<ref>Kac-Cheung, Theorem 19.1.</ref>
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| == Notes ==
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| <references/>
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| == References == | |
| *Victor Kac, Pokman Cheung, ''[[Quantum calculus|Quantum Calculus]]'', Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
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| *Jackson F H (1904), "A generalization of the functions Γ(n) and x<sub>n</sub>", ''Proc. R. Soc.'' '''74''' 64–72.
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| *Jackson F H (1910), "On q-definite integrals", ''Q. J. Pure Appl. Math.'' '''41''' 193–203.
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| [[Category:Special functions]]
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| [[Category:Q-analogs]]
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| {{mathanalysis-stub}}
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