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An extension of the operations of integration and differentiation to the case of fractional powers. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f0412301.png" /> be integrable on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f0412302.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f0412303.png" /> be the integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f0412304.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f0412305.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f0412306.png" /> is the integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f0412307.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f0412308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f0412309.png" />. One then has
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{{TEX|done}}
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An extension of the operations of integration and differentiation to the case of fractional powers. Let $f$ be integrable on the interval $[a,b]$, let $I_1^af(x)$ be the integral of $f$ along $[a,x]$, while $I_\alpha^af(x)$ is the integral of $I_{\alpha-1}^af(x)$ along $[a,b]$, $\alpha=2,3,\dots$. One then has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$I_\alpha^af(x)=\frac{1}{\Gamma(\alpha)}\int\limits_a^x(x-t)^{\alpha-1}f(t)\,dt,\quad a\leq x\leq b,\label{1}\tag{1}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123011.png" /> is the gamma-function. The right-hand side makes sense for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123012.png" />. The relation (1) defines the fractional integral (or the Riemann–Liouville integral) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123014.png" /> with starting point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123015.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123016.png" /> was studied by B. Riemann (1847) for complex values of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123017.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123018.png" /> is linear and has the semi-group property:
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where $\Gamma(\alpha)=(\alpha-1)!$ is the gamma-function. The right-hand side makes sense for every $\alpha>0$. The relation \eqref{1} defines the fractional integral (or the Riemann–Liouville integral) of order $\alpha$ of $f$ with starting point $a$. The operator $I_z^a$ was studied by B. Riemann (1847) for complex values of the parameter $z$. The operator $I_\alpha^a$ is linear and has the semi-group property:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123019.png" /></td> </tr></table>
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$$I_\alpha^a[I_\beta^af(x)]=I_{\alpha+\beta}^af(x).$$
  
The operation inverse to fractional integration is known as fractional differentiation: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123021.png" /> is the fractional derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123024.png" />, Marchaut's formula applies:
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The operation inverse to fractional integration is known as fractional differentiation: If $I_\alpha f=F$, then $f$ is the fractional derivative of order $\alpha$ of $F$. If $0<\alpha<1$, Marchaut's formula applies:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123025.png" /></td> </tr></table>
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$$f(x)=\frac{\alpha}{\Gamma(1-\alpha)}\int\limits_0^\infty\left\lbrace\frac{F(x)-F(x-t)}{t^{1+\alpha}}\right\rbrace dt.$$
  
The concept of fractional integration and differentiation was first introduced by J. Liouville (1832); he studied, in particular, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123027.png" />:
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The concept of fractional integration and differentiation was first introduced by J. Liouville (1832); he studied, in particular, the operator $I_\alpha^{-\infty}=I_\alpha$, $\alpha>0$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123028.png" /></td> </tr></table>
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$$I_\alpha f=\frac{1}{\Gamma(\alpha)}\int\limits_{-\infty}^x\frac{f(t)}{(x-t)^{1-\alpha}}dt$$
  
(subject to appropriate restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123029.png" />; cf. [[#References|[1]]], which also contains estimates of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123031.png" />).
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(subject to appropriate restrictions on $f$; cf. [[#References|[1]]], which also contains estimates of the operator $I_\alpha$ in $L_p$).
  
The following definition (H. Weyl, 1917) is convenient for an integrable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123032.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123033.png" /> with zero average value over the period. If
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The following definition (H. Weyl, 1917) is convenient for an integrable $2\pi$-periodic function $f$ with zero average value over the period. If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123034.png" /></td> </tr></table>
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$$f(x)\sim\sum_{|n|>0}c^ne^{inx}=\sum'c_ne^{inx},$$
  
then the Weyl integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123035.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123036.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123037.png" /> is defined by the formula
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then the Weyl integral $f_\alpha$ of order $\alpha>0$ of $f$ is defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$f_\alpha(x)\sim\sum'\frac{c_ne^{inx}}{(in)^\alpha};\label{2}\tag{2}$$
  
and the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123039.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123040.png" /> is defined by the equation
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and the derivative $f^\beta$ of order $\beta>0$ is defined by the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123041.png" /></td> </tr></table>
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$$f^\beta(x)=\frac{d^n}{dx^n}f_{n-\beta}(x),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123042.png" /> is the smallest integer larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123043.png" /> (it should be noted that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123044.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123045.png" />).
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where $n$ is the smallest integer larger than $\beta$ (it should be noted that $f_\alpha(x)$ coincides with $I_\alpha f(x)$).
  
 
These definitions were further developed in the framework of the theory of generalized functions. For periodic generalized functions
 
These definitions were further developed in the framework of the theory of generalized functions. For periodic generalized functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123046.png" /></td> </tr></table>
+
$$f\sim\sum'c_ne^{inx}$$
  
the operation of fractional integration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123047.png" /> is realized according to formula (2) for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123048.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123049.png" /> is negative, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123050.png" /> coincides with the partial derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123051.png" />) and has the semi-group property with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123052.png" />.
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the operation of fractional integration $I_\alpha f=f_\alpha$ is realized according to formula \eqref{2} for all real $\alpha$ (if $\alpha$ is negative, $I_\alpha f$ coincides with the partial derivative of order $\alpha$) and has the semi-group property with respect to the parameter $\alpha$.
  
In an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123053.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123054.png" /> the analogue of the operator of fractional integration is the [[Riesz potential|Riesz potential]] (or the integral of potential type)
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In an $n$-dimensional space $X$ the analogue of the operator of fractional integration is the [[Riesz potential|Riesz potential]] (or the integral of potential type)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123055.png" /></td> </tr></table>
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$$R_\alpha f(x)=\pi^{\alpha-n/2}\frac{\Gamma((n-\alpha)/2)}{\Gamma(\alpha/2)}\int\limits_X\frac{f(t)}{|x-t|^{n-\alpha}}dt.$$
  
The operation inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123056.png" /> is said to be the Riesz derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041230/f04123057.png" />.
+
The operation inverse to $R_\alpha$ is said to be the Riesz derivative of order $\alpha$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Dzhrbashyan,  "Integral transforms and representation of functions in the complex domain" , Moscow  (1966)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Dzhrbashyan,  "Integral transforms and representation of functions in the complex domain" , Moscow  (1966)  (In Russian)</TD></TR></table>

Latest revision as of 15:22, 14 February 2020

An extension of the operations of integration and differentiation to the case of fractional powers. Let $f$ be integrable on the interval $[a,b]$, let $I_1^af(x)$ be the integral of $f$ along $[a,x]$, while $I_\alpha^af(x)$ is the integral of $I_{\alpha-1}^af(x)$ along $[a,b]$, $\alpha=2,3,\dots$. One then has

$$I_\alpha^af(x)=\frac{1}{\Gamma(\alpha)}\int\limits_a^x(x-t)^{\alpha-1}f(t)\,dt,\quad a\leq x\leq b,\label{1}\tag{1}$$

where $\Gamma(\alpha)=(\alpha-1)!$ is the gamma-function. The right-hand side makes sense for every $\alpha>0$. The relation \eqref{1} defines the fractional integral (or the Riemann–Liouville integral) of order $\alpha$ of $f$ with starting point $a$. The operator $I_z^a$ was studied by B. Riemann (1847) for complex values of the parameter $z$. The operator $I_\alpha^a$ is linear and has the semi-group property:

$$I_\alpha^a[I_\beta^af(x)]=I_{\alpha+\beta}^af(x).$$

The operation inverse to fractional integration is known as fractional differentiation: If $I_\alpha f=F$, then $f$ is the fractional derivative of order $\alpha$ of $F$. If $0<\alpha<1$, Marchaut's formula applies:

$$f(x)=\frac{\alpha}{\Gamma(1-\alpha)}\int\limits_0^\infty\left\lbrace\frac{F(x)-F(x-t)}{t^{1+\alpha}}\right\rbrace dt.$$

The concept of fractional integration and differentiation was first introduced by J. Liouville (1832); he studied, in particular, the operator $I_\alpha^{-\infty}=I_\alpha$, $\alpha>0$:

$$I_\alpha f=\frac{1}{\Gamma(\alpha)}\int\limits_{-\infty}^x\frac{f(t)}{(x-t)^{1-\alpha}}dt$$

(subject to appropriate restrictions on $f$; cf. [1], which also contains estimates of the operator $I_\alpha$ in $L_p$).

The following definition (H. Weyl, 1917) is convenient for an integrable $2\pi$-periodic function $f$ with zero average value over the period. If

$$f(x)\sim\sum_{|n|>0}c^ne^{inx}=\sum'c_ne^{inx},$$

then the Weyl integral $f_\alpha$ of order $\alpha>0$ of $f$ is defined by the formula

$$f_\alpha(x)\sim\sum'\frac{c_ne^{inx}}{(in)^\alpha};\label{2}\tag{2}$$

and the derivative $f^\beta$ of order $\beta>0$ is defined by the equation

$$f^\beta(x)=\frac{d^n}{dx^n}f_{n-\beta}(x),$$

where $n$ is the smallest integer larger than $\beta$ (it should be noted that $f_\alpha(x)$ coincides with $I_\alpha f(x)$).

These definitions were further developed in the framework of the theory of generalized functions. For periodic generalized functions

$$f\sim\sum'c_ne^{inx}$$

the operation of fractional integration $I_\alpha f=f_\alpha$ is realized according to formula \eqref{2} for all real $\alpha$ (if $\alpha$ is negative, $I_\alpha f$ coincides with the partial derivative of order $\alpha$) and has the semi-group property with respect to the parameter $\alpha$.

In an $n$-dimensional space $X$ the analogue of the operator of fractional integration is the Riesz potential (or the integral of potential type)

$$R_\alpha f(x)=\pi^{\alpha-n/2}\frac{\Gamma((n-\alpha)/2)}{\Gamma(\alpha/2)}\int\limits_X\frac{f(t)}{|x-t|^{n-\alpha}}dt.$$

The operation inverse to $R_\alpha$ is said to be the Riesz derivative of order $\alpha$.

References

[1] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[2] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
[3] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[4] M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)
How to Cite This Entry:
Fractional integration and differentiation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractional_integration_and_differentiation&oldid=13622
This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article