Abstract
In previous chapters the causal and anti-causal fractional derivatives were presented. An application to shift-invariant linear systems was studied. Those derivatives were introduced into four steps: 1. Use as starting point the Grünwald–Letnikov differences and derivatives. 2. With an integral formulation for the fractional differences and using the asymptotic properties of the Gamma function obtain the generalised Cauchy derivative. 3. The computation of the integral defining the generalised Cauchy derivative is done with the Hankel path to obtain regularised fractional derivatives. 4. The application of these regularised derivatives to functions with Laplace transform, we obtain the Liouville fractional derivative and from this the Riemann–Liouville and Caputo, two-step derivatives.
Keywords
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
If 0 < b < c and \(\left|{\text{arg}}(1-z)\right|<\pi\), that function can be represented by the Euler integral: \(_2F_1(a,b;c;z)={\frac{\Upgamma (c)}{\Upgamma (b) \cdot \Upgamma (c - b)}}\int\limits_{0}^{1} {t^{b - 1} (1 - t)^{c - b - 1} (1 - zt)^{ - a} {\text{d}}t} \)
- 2.
Figure 5.2 shows the integration path and corresponding poles.
- 3.
Here we assume that ? is also non zero.
- 4.
See page 123.
- 5.
In purely mathematical terms it is a Fourier series with R b (n) as coefficients.
References
Henrici P (1991) Applied and computational complex analysis, vol 2. Wiley, New York, pp 389–391
Okikiolu GO (1966) Fourier transforms of the operator H ?. Proc Camb Philos Soc 62:73–78
Andrews GE, Askey R, Roy R (1999) Special functions. Cambridge University Press, Cambridge
Ortigueira MD (2004) From differences to differintegrations. Fract Calc Appl Anal 7(4):459–471
Ortigueira MD (2006) A coherent approach to non integer order derivatives. Signal Process Special Sect Fract Calc Appl Signals Syst 86(10):2505–2515
Ortigueira MD, Serralheiro AJ (2006) A new least-squares approach to differintegration modelling. Signal Process Special Sect Fract Calc Appl Signals Syst 86(10):2582–2591
Ortigueira MD (2006) Riesz potentials and inverses via centred derivatives. Int J Math Math Sci 2006:1–12. Article ID 48391
Samko SG, Kilbas AA, Marichev OI (1987) Fractional integrals and derivatives—theory and applications. Gordon and Breach Science Publishers, New York
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam
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 Diego
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Ortigueira, M.D. (2011). Two-Sided Fractional Derivatives. In: Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, vol 84. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0747-4_5
Download citation
DOI: https://doi.org/10.1007/978-94-007-0747-4_5
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0746-7
Online ISBN: 978-94-007-0747-4
eBook Packages: EngineeringEngineering (R0)