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.
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Notes
- 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.
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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
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