Difference between revisions of "Q-derivative"
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(Created page with "The $q$-derivative is $$D_q\{f\}(x)=\left(\dfrac{d}{dx} \right)_q f(x) = \dfrac{f(qx)-f(x)}{qx-x}.$$ =Properties= {{:q-derivative power rule}}") |
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| − | The $q$-derivative is | + | The $q$-derivative is defined by |
| − | $$ | + | $$\dfrac{\mathrm{d}_qf}{\mathrm{d}_qz}=\left\{ \begin{array}{ll} |
| + | \dfrac{f(qz)-f(z)}{(q-1)z}, & \quad z \neq 0 \\ | ||
| + | f'(0), & \quad z=0, | ||
| + | \end{array} \right.$$ | ||
| + | where $f'(0)$ denotes the [[derivative]]. | ||
=Properties= | =Properties= | ||
| − | {{ | + | [[Relationship between q-derivative and derivative]]<br /> |
| + | [[q-derivative power rule]]<br /> | ||
| + | |||
| + | =References= | ||
| + | * {{PaperReference|q-exponential and q-gamma functions. I. q-exponential functions|1994|D.S. McAnally|prev=findme|next=Q-derivative power rule}} $(2.1)$ | ||
| + | * {{BookReference|Quantum Calculus|2002|Victor Kac|author2=Pokman Cheung||prev=findme|next=findme}} $(1.5)$ | ||
Latest revision as of 04:05, 26 December 2016
The $q$-derivative is defined by $$\dfrac{\mathrm{d}_qf}{\mathrm{d}_qz}=\left\{ \begin{array}{ll} \dfrac{f(qz)-f(z)}{(q-1)z}, & \quad z \neq 0 \\ f'(0), & \quad z=0, \end{array} \right.$$ where $f'(0)$ denotes the derivative.
Properties
Relationship between q-derivative and derivative
q-derivative power rule
References
- D.S. McAnally: q-exponential and q-gamma functions. I. q-exponential functions (1994)... (previous)... (next) $(2.1)$
- 2002: Victor Kac and Pokman Cheung: Quantum Calculus ... (previous) ... (next) $(1.5)$
