The q-derivative and q-integral
The q-derivative and q-integral
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \index{q-Derivative operator@$q$-Derivative operator} \qderiv{q}f(z):=\left\{\begin{array}{ll} \displaystyle \frac{f(z)-f(qz)}{(1-q)z}, & z\neq 0\\[5mm] f'(0), & z=0.\end{array}\right. }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv[0]{q}f:=f\quad\textrm{and}\quad \qderiv[n]{q}f:=\qderiv{q}@{\qderiv[n-1]{q}f} \quad n=1,2,3,\ldots }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{q\rightarrow 1}\qderiv{q}f(z)=f'(z) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}@{f(\gamma x)}=\gamma\left(\qderiv{q}f\right)(\gamma x) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv[n]{q}@{f(\gamma x)}=\gamma^n\left(\qderiv[n]{q}f\right)(\gamma x) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}@{f(x)g(x)}= f(qx)\qderiv{q}g(x)+g(x)\qderiv{q}f(x) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv[n]{q}@{f(x)g(x)}=\sum_{k=0}^n\qBinomial{n}{k}{q} \left(\qderiv[n-k]{q}f\right)(q^kx)\left(\qderiv[k]{q}g\right)(x) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \index{q-Integral@$q$-Integral}\index{Jackson-Thomae q-integral@Jackson-Thomae $q$-integral} \int_0^zf(t)\,d_qt:=z(1-q)\sum_{n=0}^{\infty}f(q^nz)q^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_0^{\infty}f(t)\,d_qt:=(1-q)\sum_{n=-\infty}^{\infty}f(q^n)q^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{q\rightarrow 1}\int_0^zf(t)\,d_qt=\int_0^zf(t)\,dt }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_a^bf(t)\,d_qt=b(1-q)\sum_{n=0}^{\infty}f(bq^n)q^n-a(1-q)\sum_{n=0}^{\infty}f(aq^n)q^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-\infty}^{\infty}f(t)\,d_qt=(1-q)\sum_{n=-\infty}^{\infty}\left\{f(q^n)+f(-q^n)\right\}q^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle F(z)=\int_0^zf(t)\,d_qt\quad\Longrightarrow\quad\qderiv{q}F(z)=f(z) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_a^b\qderiv{q}f(t)\,d_qt=f(b)-f(a) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_a^bg(x)\qderiv{q}f(x)\,d_qx=\Big[f(x)g(x)\Big]_a^b-\int_a^bf(qx)\qderiv{q}g(x)\,d_qx }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_a^b\frac{\qPochhammer{a^{-1}qt,b^{-1}qt,ct}{q}{\infty}}{\qPochhammer{dt,et,ft}{q}{\infty}}\,d_qt {}=(b-a)(1-q)\,\frac{\qPochhammer{q,a^{-1}bq,ab^{-1}q,cd^{-1},c\expe^{-1},cf^{-1}}{q}{\infty}} {\qPochhammer{ad,ae,af,bd,be,bf}{q}{\infty}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_a^b\frac{\qPochhammer{a^{-1}qt,b^{-1}qt}{q}{\infty}}{\qPochhammer{dt,et}{q}{\infty}}\,d_qt =(b-a)(1-q)\,\frac{\qPochhammer{q,a^{-1}bq,ab^{-1}q,abde}{q}{\infty}}{\qPochhammer{ad,ae,bd,be}{q}{\infty}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_a^{\infty}\frac{\qPochhammer{a^{-1}qt}{q}{\infty}}{\qPochhammer{dt,et}{q}{\infty}}\,d_qt =(1-q)\,\frac{\qPochhammer{q,a,a^{-1}q,ade,a^{-1}d^{-1}\expe^{-1}q}{q}{\infty}} {\qPochhammer{ad,ae,d,d^{-1}q,e,\expe^{-1}q}{q}{\infty}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-\infty}^{\infty}\frac{1}{\qPochhammer{dt,et}{q}{\infty}}\,d_qt {}=(1-q)\,\frac{\qPochhammer{q,-q,-1,-de,-d^{-1}\expe^{-1}q}{q}{\infty}} {\qPochhammer{d,-d,d^{-1}q,-d^{-1}q,e,-e,\expe^{-1}q,-\expe^{-1}q}{q}{\infty}} }}