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| '''Quantum calculus''', sometimes called '''calculus without limits''', is equivalent to traditional [[infinitesimal calculus]] without the notion of [[Limit of a function|limits]]. It defines "q-calculus" and "h-calculus". h ostensibly stands for [[Planck's constant]] while ''q'' stands for quantum. The two parameters are related by the formula
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| :<math>q = e^{i h} = e^{2 \pi i \hbar} \,</math>
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| where <math>\scriptstyle \hbar = \frac{h}{2 \pi} \,</math> is the [[reduced Planck constant]].
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| ==Differentiation==
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| In the q-calculus and h-calculus, [[differential of a function|differentials]] of functions are defined as
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| :<math>d_q(f(x)) = f(qx) - f(x) \,</math>
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| and
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| :<math>d_h(f(x)) = f(x + h) - f(x) \,</math>
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| respectively. [[Derivative]]s of functions are then defined as fractions by the [[q-derivative]]
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| :<math>D_q(f(x)) = \frac{d_q(f(x))}{d_q(x)} = \frac{f(qx) - f(x)}{(q - 1)x}</math>
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| and by
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| :<math>D_h(f(x)) = \frac{d_h(f(x))}{d_h(x)} = \frac{f(x + h) - f(x)}{h}</math>
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| In the [[Limit of a function|limit]], as h goes to 0, or equivalently as q goes to 1, these expressions take on the form of the derivative of classical calculus.
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| ==Integration==
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| ===q-integral===
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| A function F(x) is a q-antiderivative of f(x) if D<sub>q</sub>F(x)=f(x). The q-antiderivative (or q-integral) is denoted by <math>\int f(x)d_qx</math> and an expression for F(x) can be found from the formula
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| <math>\int f(x)d_qx = (1-q)\sum_{j=0}^\infty xq^jf(xq^j)</math> which is called the [[Jackson integral]] of f(x). For 0 < q < 1, the series converges to a function F(x) on an interval (0,A] if |f(x)x^α| is bounded on the interval (0,A] for some 0 <= α < 1.
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| The q-integral is a [[Riemann-Stieltjes integral]] with respect to a [[step function]] having infinitely many points of increase at the points q<sup>j</sup>, with the jump at the point q<sup>j</sup> being q<sup>j</sup>. If we call this step function g<sub>q</sub>(t) then dg<sub>q</sub>(t) = d<sub>q</sub>t.<ref>[http://www.mat.uc.pt/preprints/ps/p0432.pdf FUNCTIONS q-ORTHOGONAL WITH RESPECT TO THEIR OWN ZEROS], LUIS DANIEL ABREU, Pre-Publicacoes do Departamento de Matematica Universidade de Coimbra, Preprint Number 04–32</ref>
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| ===h-integral===
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| A function F(x) is an h-antiderivative of f(x) if D<sub>h</sub>F(x)=f(x). The h-antiderivative (or h-integral) is denoted by <math>\int f(x)d_hx</math>. If a and b differ by an integer multiple of h then the definite integral<math>\int_a^b f(x)d_hx</math> is given by a [[Riemann sum]] of f(x) on the interval [a,b] partitioned into subintervals of width h.
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| == Example ==
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| The derivative of the function <math>x^n</math> (for some positive integer <math>n</math>) in the classical calculus is <math>nx^{n-1}</math>. The corresponding expressions in q-calculus and h-calculus are
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| :<math>D_q(x^n) = \frac{q^n - 1}{q - 1} x^{n - 1} = [n]_q\ x^{n - 1}</math>
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| with the [[q-bracket]]
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| :<math>[n]_q = \frac{q^n - 1}{q - 1}</math>
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| and
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| :<math>D_h(x^n) = x^{n - 1} + h x^{n - 2} + \cdots + h^{n - 1}</math>
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| respectively. The expression <math>[n]_q x^{n - 1}</math> is then the q-calculus analogue of the simple power rule for
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| positive integral powers. In this sense, the function <math>x^n</math> is still ''nice'' in the q-calculus, but rather
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| ugly in the h-calculus – the h-calculus analog of <math>x^n</math> is instead the [[falling factorial]], <math>(x)_n := x(x-1)\cdots(x-n+1).</math>
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| One may proceed further and develop, for example, equivalent notions of [[Q-derivative|Taylor expansion]], et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the [[sine]] function whose q-derivative is the appropriate analogue for the [[cosine]].
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| == History ==
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| The h-calculus is just the [[calculus of finite differences]], which had been studied by [[George Boole]] and others, and has proven useful in a number of fields, among them [[combinatorics]] and [[fluid mechanics]]. The q-calculus, while dating in a sense back to [[Leonhard Euler]] and [[Carl Gustav Jacobi]], is only recently beginning to see more usefulness in [[quantum mechanics]], having an intimate connection with commutativity relations and [[Lie algebra]].
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| == See also ==
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| * [[Noncommutative geometry]]
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| * [[Quantum differential calculus]]
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| * [[Time scale calculus]]
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| * [[q-analog]]
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| ==References==
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| {{reflist}}
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| <!-- this section is for *references*, sources used to write a part of the article or cited in the article to justify a statement. Supplementary reading should go into "further reading" -->
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| * F. H. Jackson (1908), "On q-functions and a certain difference operator", ''Trans. Roy. Soc. Edin.'', '''46''' 253-281.
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| * Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914 , ISBN 0470274530 , ISBN 978-0470274538
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| *[[Victor Kac]], [[Pokman Cheung]], Quantum calculus'', Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
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| [[Category:Quantum mechanics| ]]
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| [[Category:Differential calculus| ]]
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