Q導數也稱為傑克遜導數,乃是一般導數的Q模擬,由英國數學家F. H. Jackson創立。
定義
函數f(x)的q-導數定義如下:
![{\displaystyle \left({\frac {d}{dx}}\right)_{q}f(x)={\frac {f(qx)-f(x)}{qx-x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09a128672687b4e77883385aed3f9b09611df21f)
或書寫為
.
![{\displaystyle D_{q}={\frac {1}{x}}~{\frac {q^{d~~~ \over d(\ln x)}-1}{q-1}}~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb90e565a6fc1d9dfccbf09a40fec25cbe4f158)
當as q → 1時,化為尋常的導數, → d⁄dx,
關係式
q-導數算符是一個線性算子:
![{\displaystyle \displaystyle D_{q}(f(x)+g(x))=D_{q}f(x)+D_{q}g(x)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9448de46611d96624e577bf9ac08219c4f30d582)
![{\displaystyle \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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7093c2943b5a855c2466fca6eeedbd8df2f27a93)
![{\displaystyle \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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/295f61139703299586d5343d6005bcc1a5ad2fab)
若
. 則
![{\displaystyle \displaystyle D_{q}f(g(x))=D_{q^{k}}(f)(g(x))D_{q}(g)(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88473537b0a6ab7bab4ba2da7904c3a381dfb9be)
q-導數 的本徵值是q-指數 eq(x).
與導數的關係
![{\displaystyle \left({\frac {d}{dz}}\right)_{q}z^{n}={\frac {1-q^{n}}{1-q}}z^{n-1}=[n]_{q}z^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64a0fdcac5de2e43d295dde69328b13b64d51c6c)
其中
是n的 q括號
並且
.
一個函數的n階導數為:
![{\displaystyle (D_{q}^{n}f)(0)={\frac {f^{(n)}(0)}{n!}}{\frac {(q;q)_{n}}{(1-q)^{n}}}={\frac {f^{(n)}(0)}{n!}}[n]_{q}!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffbbfd8c17009de664036b8758cb8dcc679638a)
![{\displaystyle 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}!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb3f1e9b829520538feb6987126b00f1c1cc6c53)
例子
q derivative of sin(x)
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q derivative of sin(x) 3D plot
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q derivative of sin(x) 2D animation
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q derivative of sin(x) density plot
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q derivative of tanh(x) animation
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q derivative of tanh(x) 3D
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q derivative of tanh(z) complex 3D
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q derivative of tanh(z) 2D density
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參見
參考文獻
- F. H. Jackson (1908), On q-functions and a certain difference operator, Trans. Roy. Soc. Edin., 46 253-281.
- Victor Kac, Pokman Cheung, Quantum Calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
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