查看“︁Q导数”︁的源代码

'''Q导数'''也称为杰克逊导数，乃是一般[[导数]]的[[Q模拟]]，由[[英国]][[数学家]]{{le|F. H. Jackson}}创立。
==定义==
函数''f''(''x'')的q-导数定义如下：

:<math>\left(\frac{d}{dx}\right)_q f(x)=\frac{f(qx)-f(x)}{qx-x}.</math>

或书写为 <math>D_qf(x)</math>. 


:<math>D_q= \frac{1}{x} ~ \frac{q^{d~~~ \over d (\ln x)}  -1}{q-1} ~, </math>
当as ''q'' → 1时，化为寻常的导数, → <sup>''d''</sup>&frasl;<sub>''dx''</sub>,  


==关系式==
q-导数算符是一个线性算子：
:<math>\displaystyle D_q (f(x)+g(x)) = D_q f(x) + D_q g(x)~.</math>


:<math>\displaystyle D_q (f(x)g(x)) = g(x)D_q f(x) + f(qx)D_q g(x) = g(qx)D_q f(x) + f(x)D_q g(x). </math>



:<math>\displaystyle D_q (f(x)/g(x)) = \frac{g(x)D_q f(x) - f(x)D_q g(x)}{g(qx)g(x)},\quad g(x)g(qx)\neq 0. </math>

若 <math>g(x) = c x^k</math>. 则

:<math>\displaystyle D_q f(g(x)) = D_{q^k}(f)(g(x))D_q(g)(x).</math>

''q''-导数 的[[本征值]]是[[q-指数]] ''e<sub>q</sub>''(''x'').

==与导数的关系==

:<math>\left(\frac{d}{dz}\right)_q z^n = \frac{1-q^n}{1-q} z^{n-1} = 
[n]_q z^{n-1}</math>

其中 <math>[n]_q</math> 是n的 [[q括号]]  

并且 <math>\lim_{q\to 1}[n]_q = n</math> .

一个函数的n阶导数为：

:<math>(D^n_q f)(0)=
\frac{f^{(n)}(0)}{n!} \frac{(q;q)_n}{(1-q)^n}= 
\frac{f^{(n)}(0)}{n!} [n]_q!
</math>


:<math>f(z)=\sum_{n=0}^\infty f^{(n)}(0)\,\frac{z^n}{n!} = \sum_{n=0}^\infty (D^n_q f)(0)\,\frac{z^n}{[n]_q!}</math>

==例子==

<math>D_{q}sin(x)={\frac {\sin \left( qx \right) -\sin \left( x \right) }{ \left( q-1
 \right) x}}</math>
{|
|[[File:Dsin1.gif|thumb|q derivative of  sin(x)]]
|[[File:Dsin2.gif|thumb|q derivative of  sin(x) 3D plot]]
|[[File:Dsin5.gif|thumb|q derivative of  sin(x)  2D  animation]]
|[[File:Dsin3.JPG|thumb|q derivative of  sin(x) density plot]]
|}

<math>D_{q}tanh(x)={\frac {\tanh \left( qx \right) -\tanh \left( x \right) }{ \left( q-1
 \right) x}}
</math>
{|
|[[File:Dtanh2.gif|thumb|q derivative of  tanh(x)  animation]]
|[[File:Dtanh.png|thumb|q derivative of  tanh(x) 3D]]
|[[File:Dtanh3.gif|thumb|q derivative of  tanh(z) complex  3D]]
|[[File:Dtanh4.gif|thumb|q derivative of  tanh(z)   2D  density]]
|}

== 参见 ==
* {{tsl|en|Derivative (generalizations)}}
* {{tsl|en|Jackson integral|杰克逊积分}}
* [[Q指数]]
* {{tsl|en|Q-difference polynomial|Q-差值积分}}
* {{tsl|en|Quantum calculus|量子微积分}}
* {{tsl|en|Tsallis entropy}}

==参考文献==
* 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

==延伸阅读==
* J. Koekoek, R. Koekoek, ''[http://arxiv.org/abs/math/9908140 A note on the q-derivative operator]{{Wayback|url=http://arxiv.org/abs/math/9908140 |date=20190217161750 }}'', (1999) ArXiv math/9908140
* Thomas Ernst, ''[https://web.archive.org/web/20150824041046/http://www2.math.uu.se/research/pub/Ernst4.pdf The History of q-Calculus and a new method]'',(2001),

{{q超几何函数}}
[[Category:微分学]]
[[Category:Q-模拟]]