Integral Representations

Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 84)


In the previous chapter we addressed the problem of fractional derivative definition and proposed the use the Grünwald–Letnikov and in particular the forward and backward derivatives. These choices were motivated by five main reasons they: do not need superfluous derivative computations, do not insert unwanted initial conditions, are more flexible, allow sequential computations, are more general in the sense of allowing to be applied to a large class of functions.


Backward Derivative Fractional Derivative Definition Grunwald Letnikov (GL) Hankel Contour Negative Real Half-axis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Diaz JB, Osler TJ (1974) Differences of fractional order. Math Comput 28(125):185–202MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kiryakova V (1996) A long standing conjecture failed? In: Proceedings of the 2nd international workshop on transform methods & special functions, Varna’96 Aug 1996Google Scholar
  3. 3.
    Ortigueira MD (2004) From differences to differintegrations. Fract Calc Appl Anal 7(4):459–471MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ortigueira MD (2006) A coherent approach to non integer order derivatives. Signal Process, Special Section Fractional Calculus Applications in Signals and Systems. 86(10):2505–2515Google Scholar
  5. 5.
    Samko SG, Kilbas AA, Marichev OI (1987) Fractional integrals and derivatives—theory and applications. Gordon and Breach Science Publishers, New YorkGoogle Scholar
  6. 6.
    Nishimoto K (1989) Fractional calculus. Descartes Press Co., KoriyamazbMATHGoogle Scholar
  7. 7.
    Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkzbMATHGoogle Scholar
  8. 8.
    Podlubny I (1999) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press, San DiegozbMATHGoogle Scholar
  9. 9.
    Hoskins RF, Pinto JS (1994) Distributions ultradistributions and other generalised functions. Ellis Horwood Limited, ChichesterzbMATHGoogle Scholar
  10. 10.
    Zemanian AH (1987) Distribution theory and transform analysis. Dover Publications, New YorkzbMATHGoogle Scholar
  11. 11.
    Poularikas AD (ed) (2000) The transforms and applications handbook. CRC Press, Boca RatonGoogle Scholar
  12. 12.
    Chaudhry MA, Zubair SM (2002) On a class of incomplete gamma functions with applications. Chapman Hall, LondonGoogle Scholar
  13. 13.
    Henrici P (1991) Applied and computational complex analysis, vol 2. Wiley, New York, pp 389–391Google Scholar
  14. 14.
    Ortigueira MD (2005) Two new integral formulae for the Beta function. Int J Appl Math 18(1):109–116MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Faculdade de Ciências/Tecnologia da UNLUNINOVA and DEECaparicaPortugal

Personalised recommendations