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:''Fractional integration redirects here. Not to be confused with Autoregressive fractionally integrated moving average In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by : is the fractional derivative (if ''q'' > 0) or fractional integral (if ''q'' < 0). If ''q'' = 0, then the ''q''-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral. ==Standard definitions== The three most common forms are: *The Riemann–Liouville differintegral :This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the Cauchy formula for repeated integration to arbitrary order. : *The Grunwald–Letnikov differintegral :The Grunwald–Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot. : *The Weyl differintegral :This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「:'''''Fractional integration''' redirects here. Not to be confused with Autoregressive fractionally integrated moving averageIn fractional calculus, an area of applied mathematics, the '''differintegral''' is a combined differentiation/integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by:\mathbb^qfis the fractional derivative (if ''q'' > 0) or fractional integral (if ''q'' ==Standard definitions==The three most common forms are:*The Riemann–Liouville differintegral:This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the Cauchy formula for repeated integration to arbitrary order.: \begin \\& =\frac \frac \int_^t (t-\tau)^f(\tau)d\tau\end*The Grunwald–Letnikov differintegral:The Grunwald–Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot.: \begin \\& =\lim_\left()^\sum_^(-1)^jf\left(t-j\left()\right)\end*The Weyl differintegral:This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period.」の詳細全文を読む :''Fractional integration redirects here. Not to be confused with Autoregressive fractionally integrated moving average In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by : is the fractional derivative (if ''q'' > 0) or fractional integral (if ''q'' < 0). If ''q'' = 0, then the ''q''-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral. ==Standard definitions== The three most common forms are: *The Riemann–Liouville differintegral :This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the Cauchy formula for repeated integration to arbitrary order. : *The Grunwald–Letnikov differintegral :The Grunwald–Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot. : *The Weyl differintegral :This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「:''Fractional integration redirects here. Not to be confused with Autoregressive fractionally integrated moving averageIn fractional calculus, an area of applied mathematics, the differintegral''' is a combined differentiation/integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by:\mathbb^qfis the fractional derivative (if ''q'' > 0) or fractional integral (if ''q'' ==Standard definitions==The three most common forms are:*The Riemann–Liouville differintegral:This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the Cauchy formula for repeated integration to arbitrary order.: \begin \\& =\frac \frac \int_^t (t-\tau)^f(\tau)d\tau\end*The Grunwald–Letnikov differintegral:The Grunwald–Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot.: \begin \\& =\lim_\left()^\sum_^(-1)^jf\left(t-j\left()\right)\end*The Weyl differintegral:This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period.」の詳細全文を読む differintegral''' is a combined differentiation/integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by:\mathbb^qfis the fractional derivative (if ''q'' > 0) or fractional integral (if ''q'' ==Standard definitions==The three most common forms are:*The Riemann–Liouville differintegral:This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the Cauchy formula for repeated integration to arbitrary order.: \begin \\& =\frac \frac \int_^t (t-\tau)^f(\tau)d\tau\end*The Grunwald–Letnikov differintegral:The Grunwald–Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot.: \begin \\& =\lim_\left()^\sum_^(-1)^jf\left(t-j\left()\right)\end*The Weyl differintegral:This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period.」の詳細全文を読む スポンサード リンク
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