Quantum calculus
Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while "q" stands for quantum. The two parameters are related by the formula where formula_2 is the reduced Planck constant. Differentiation. In the q-calculus and h-calculus, differentials of functions are defined as and respectively. Derivatives of functions are then defined as fractions by the q-derivative and by In the 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. Integration. q-integral. A function "F"("x") is a q-antiderivative of "f"("x") if "D""F"("x") = "f"("x"). The q-antiderivative (or q-integral) is denoted by formula_7 and an expression for "F"("x") can be found from the formula formula_8 which is called the Jackson integral of "f"("x"). For , the series converges to a function "F"("x") on an interval (0,"A"] if |"f"("x")"x"| is bounded on the interval for some . The q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at  the points "q", with the jump at the point "q" being "q". If we call this step function "g"("t") then "dg"("t") = "d""t". h-integral. A function "F"("x") is an h-antiderivative of "f"("x") if "D""F"("x") = "f"("x"). The h-antiderivative (or h-integral) is denoted by formula_9. If "a" and "b" differ by an integer multiple of "h" then the definite integral formula_10 is given by a Riemann sum of "f"("x") on the interval partitioned into subintervals of width "h". Example. The derivative of the function formula_11 (for some positive integer formula_12) in the classical calculus is formula_13. The  corresponding expressions in q-calculus and h-calculus are with the q-bracket and respectively. The expression formula_17 is then the q-calculus analogue of the simple power rule for positive integral powers. In this sense, the function formula_11 is still "nice" in the q-calculus, but rather ugly in the h-calculus – the h-calculus analog of formula_11 is instead the falling factorial, formula_20 One may proceed further and develop, for example, equivalent notions of 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. History. 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.