Difference between revisions of "Q-derivative"

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The $q$-derivative is  
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The $q$-derivative is defined by
$$\left(\dfrac{d}{dx} \right)_q f(x) =D_q\{f\}(x)=\left\{ \begin{array}{ll}
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$$\dfrac{\mathrm{d}_qf}{\mathrm{d}_qz}=\left\{ \begin{array}{ll}
\dfrac{f(qx)-f(x)}{qx-x} &; x \neq 0 \\
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\dfrac{f(qz)-f(z)}{(q-1)z}, & \quad z \neq 0 \\
f'(0) &; x=0.
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f'(0), & \quad z=0,
 
\end{array} \right.$$
 
\end{array} \right.$$
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where $f'(0)$ denotes the [[derivative]].
  
 
=Properties=
 
=Properties=
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=References=
 
=References=
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* {{PaperReference|q-exponential and q-gamma functions. I. q-exponential functions|1994|D.S. McAnally|prev=findme|next=Q-derivative power rule}} $(2.1)$
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* {{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