Quantum calculus

Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. The two types of calculus in quantum calculus are q-calculus and h-calculus. The goal of both types is to find "analogs" of mathematical objects, where, after taking a certain limit, the original object is returned. In q-calculus, the limit as q tends to 1 is taken of the q-analog. Likewise, in h-calculus, the limit as h tends to 0 is taken of the h-analog. The parameters ${\displaystyle q}$ and ${\displaystyle h}$ can be related by the formula ${\displaystyle q=e^{h}}$.

The q-differential and h-differential are defined as:

and

respectively. The q-derivative and h-derivative are then defined as

and

respectively. By taking the limit as ${\displaystyle q\rightarrow 1}$ of the q-derivative or as ${\displaystyle h\rightarrow 0}$ of the h-derivative, one can obtain the derivative:

A function F(x) is a q-antiderivative of f(x) if D_qF(x) = f(x). The q-antiderivative (or q-integral) is denoted by ${\textstyle \int f(x)\,d_{q}x}$ and an expression for F(x) can be found from:${\textstyle \int f(x)\,d_{q}x=(1-q)\sum _{j=0}^{\infty }xq^{j}f(xq^{j})}$, 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.

The q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points q^j..The jump at the point q^j is q^j. Calling this step function g_q(t) gives dg_q(t) = d_qt.