Wet-bulb temperature

Template:Lowercase In mathematics, in the area of combinatorics, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration

Definition

The q-derivative of a function f(x) is defined as

{(\frac{d}{d x})}_{q} f (x) = \frac{f (q x) - f (x)}{q x - x} .

It is also often written as $D_{q} f (x)$ . The q-derivative is also known as the Jackson derivative.

Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator

D_{q} = \frac{1}{x} \frac{q^{\frac{d}{d (\ln x)}} - 1}{q - 1},

which goes to the plain derivative, →^d⁄_dx, as q→1.

It is manifestly linear,

D_{q} (f (x) + g (x)) = D_{q} f (x) + D_{q} g (x) .

It has product rule analogous to the ordinary derivative product rule, with two equivalent forms

D_{q} (f (x) g (x)) = g (x) D_{q} f (x) + f (q x) D_{q} g (x) = g (q x) D_{q} f (x) + f (x) D_{q} g (x) .

Similarly, it satisfies a quotient rule,

D_{q} (f (x) / g (x)) = \frac{g (x) D_{q} f (x) - f (x) D_{q} g (x)}{g (q x) g (x)}, g (x) g (q x) \neq 0 .

There is also a rule similar to the chain rule for ordinary derivatives. Let $g (x) = c x^{k}$ . Then

D_{q} f (g (x)) = D_{q^{k}} (f) (g (x)) D_{q} (g) (x) .

The eigenfunction of the q-derivative is the q-exponential e_q(x).

Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:

{(\frac{d}{d z})}_{q} z^{n} = \frac{1 - q^{n}}{1 - q} z^{n - 1} = [n]_{q} z^{n - 1}

where $[n]_{q}$ is the q-bracket of n. Note that $\lim_{q \to 1} [n]_{q} = n$ so the ordinary derivative is regained in this limit.

The n-th q-derivative of a function may be given as:

(D_{q}^{n} f) (0) = \frac{f^{(n)} (0)}{n!} \frac{(q; q)_{n}}{(1 - q)^{n}} = \frac{f^{(n)} (0)}{n!} [n]_{q}!

provided that the ordinary n-th derivative of f exists at x=0. Here, $(q; q)_{n}$ is the q-Pochhammer symbol, and $[n]_{q}!$ is the q-factorial. If $f (x)$ is analytic we can apply the Taylor formula to the definition of $D_{q} (f (x))$ to get

D_{q} (f (x)) = \sum_{k = 0}^{\infty} \frac{(q - 1)^{k}}{(k + 1)!} x^{k} f^{(k + 1)} (x) .

A q-analog of the Taylor expansion of a function about zero follows:

f (z) = \sum_{n = 0}^{\infty} f^{(n)} (0) \frac{z^{n}}{n!} = \sum_{n = 0}^{\infty} (D_{q}^{n} f) (0) \frac{z^{n}}{[n]_{q}!}

F. H. Jackson (1908), "On q-functions and a certain difference operator", Trans. Roy. Soc. Edin., 46 253-281.

Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538

Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8