Difference between revisions of "Q-derivative"
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The $q$-derivative is | The $q$-derivative is | ||
$$\left(\dfrac{d}{dx} \right)_q f(x) =D_q\{f\}(x)=\left\{ \begin{array}{ll} | $$\left(\dfrac{d}{dx} \right)_q f(x) =D_q\{f\}(x)=\left\{ \begin{array}{ll} | ||
| − | \dfrac{f(qx)-f(x)}{qx-x} &; | + | \dfrac{f(qx)-f(x)}{qx-x} &; x \neq 0 \\ |
| − | f'(0) &; | + | f'(0) &; x=0. |
\end{array} \right.$$ | \end{array} \right.$$ | ||
=Properties= | =Properties= | ||
{{:q-derivative power rule}} | {{:q-derivative power rule}} | ||
Revision as of 00:32, 4 May 2015
The $q$-derivative is $$\left(\dfrac{d}{dx} \right)_q f(x) =D_q\{f\}(x)=\left\{ \begin{array}{ll} \dfrac{f(qx)-f(x)}{qx-x} &; x \neq 0 \\ f'(0) &; x=0. \end{array} \right.$$
Contents
Properties
Theorem
The following formula holds: $$D_q(z^n)=[n]_q z^{n-1},$$ where $D_q$ denotes the $q$-derivative and $[n]_q$ denotes the $q$-number.
