Q导数

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F. H. Jackson创立。

定义

函数f(x)的q-导数定义如下：

$\left({\frac {d}{dx}}\right)_{q}f(x)={\frac {f(qx)-f(x)}{qx-x}}.$

或书写为 $D_{q}f(x)$.

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

当as q → 1时，化为寻常的导数, → ^d⁄_dx,

关系式

q-导数算符是一个线性算子：

$\displaystyle D_{q}(f(x)+g(x))=D_{q}f(x)+D_{q}g(x)~.$

$\displaystyle D_{q}(f(x)g(x))=g(x)D_{q}f(x)+f(qx)D_{q}g(x)=g(qx)D_{q}f(x)+f(x)D_{q}g(x).$

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

若 $g(x)=cx^{k}$. 则

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

q-导数 的本征值是q-指数 e_q(x).

与导数的关系

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

其中 $[n]_{q}$ 是n的 q括号

并且 $\lim _{q\to 1}[n]_{q}=n$ .

一个函数的n阶导数为：

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

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

例子

$D_{q}sin(x)={\frac {\sin \left(qx\right)-\sin \left(x\right)}{\left(q-1\right)x}}$

q derivative of sin(x)

q derivative of sin(x) 3D plot

q derivative of sin(x) 2D animation

q derivative of sin(x) density plot

$D_{q}tanh(x)={\frac {\tanh \left(qx\right)-\tanh \left(x\right)}{\left(q-1\right)x}}$

q derivative of tanh(x) animation

q derivative of tanh(x) 3D

q derivative of tanh(z) complex 3D

q derivative of tanh(z) 2D density

参见

• Derivative (generalizations)
• 杰克逊积分
• Q指数
• Q-差值积分
• 量子微积分
• 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, A note on the q-derivative operator（页面存档备份，存于互联网档案馆）, (1999) ArXiv math/9908140
• Thomas Ernst, The History of q-Calculus and a new method,(2001),