GV

GeezerVison

The World Wide Web, restored to 800 pixels of glory.

Home|About|Gaming cabinet|Archives|How it works|Starter sites

On this site

  • Algebra
  • Applied Mathematics
  • Calculus and Analysis
  • Discrete Mathematics
  • Foundations of Mathematics
  • Geometry
  • History and Terminology
  • Number Theory
  • Probability and Statistics
  • Recreational Mathematics
  • Topology
  • Alphabetical Index
  • Random Entry
  • New in MathWorld

This view

  • Original page
  • New address

GeezerVison is showing mathworld.wolfram.com — Fractional Integral -- from Wolfram MathWorld

Search

Fractional Integral

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The fractional integral extends the repeated integral from integer to noninteger order. Denote the nth derivative D^n and the n-fold integral D^(-n). Then

D^(-1)f(t)=int_0^tf(xi)dxi.
(1)

Now, if the equation

D^(-n)f(t)=1/((n-1)!)int_0^t(t-xi)^(n-1)f(xi)dxi
(2)

for the multiple integral is true for n, then

D^(-(n+1))f(t)=D^(-1)[1/((n-1)!)int_0^t(t-xi)^(n-1)f(xi)dxi]
(3)
=int_0^t[1/((n-1)!)int_0^x(x-xi)^(n-1)f(xi)dxi]dx.
(4)

Interchanging the order of integration gives

D^(-(n+1))f(t)=1/(n!)int_0^t(t-xi)^nf(xi)dxi.
(5)

But (3) is true for n=1, so it is also true for all n by mathematical induction. The fractional integral of f(t) of order nu>0 can then be defined by

D^(-nu)f(t)=1/(Gamma(nu))int_0^t(t-xi)^(nu-1)f(xi)dxi,
(6)

where Gamma(nu) is the gamma function.

More generally, the Riemann-Liouville operator of fractional integration is defined as

_aD_t^(-nu)f(t)=1/(Gamma(nu))int_a^t(t-xi)^(nu-1)f(xi)dxi
(7)

for nu>0 with _aD_t^0f(t)=f(t) (Oldham and Spanier 1974, Miller and Ross 1993, Srivastava and Saxena 2001, Saxena 2002).

The fractional integral of order 1/2 is called a semi-integral. In an Abel integral equation, the given function is a scalar multiple of a fractional integral of the unknown function.

Few functions have a fractional integral expressible in terms of elementary functions. Exceptions include

D^(-nu)t^lambda=(Gamma(lambda+1))/(Gamma(lambda+nu+1))t^(lambda+nu) for lambda>-1,nu>0
(8)
D^(-nu)e^(at)=1/(Gamma(nu))e^(at)int_0^tx^(nu-1)e^(-ax)dx
(9)
=(a^(-nu)e^(at)gamma(nu,at))/(Gamma(nu))
(10)
=E_t(nu,a),
(11)

where gamma(a,x) is a lower incomplete gamma function and E_t(nu,a) is the Et-function. From (10), the fractional integral of the constant function f(t)=c is given by

D^(-nu)c=clim_(lambda->0)(Gamma(lambda+1))/(Gamma(lambda+nu+1))t^(lambda+nu)
(12)
=c(t^nu)/(Gamma(nu+1)).
(13)

A fractional derivative can also be similarly defined. The study of fractional derivatives and integrals is called fractional calculus.

See also

Fractional Calculus, Riemann-Liouville Fractional Derivative, Riemann-Liouville Operator, Semi-Integral

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

  • 2x^2 - 3xy + 4y^2 + 6x - 3y - 4 = 0
  • continued fraction tan x
  • Menger sponge

References

Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993.Oldham, K. B. and Spanier, J. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press, 1974.Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, 1993.Saxena, R. K.; Mathai, A. M.; and Haubold, H. J. "On Fractional Kinetic Equations." 23 Jun 2002. https://arxiv.org/abs/math/0206240.Srivastava, H. M. and Saxena, R. K. "Operators of Fractional Integration and Their Applications." Appl. Math. and Comput. 118, 1-52, 2001.

Referenced on Wolfram|Alpha

Fractional Integral

Cite this as:

Weisstein, Eric W. "Fractional Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FractionalIntegral.html

Subject classifications

You are visitor 01402503 — GeezerVison does not store the pages you look up.
Best viewed with Netscape Navigator 4.7 or Internet Explorer 6 at 800×600. No cookies required.