By J. Sabatier, J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado

In the final twenty years, fractional (or non integer) differentiation has performed a crucial position in a variety of fields akin to mechanics, electrical energy, chemistry, biology, economics, keep an eye on idea and sign and photograph processing. for instance, within the final 3 fields, a few vital concerns similar to modelling, curve becoming, filtering, trend attractiveness, part detection, identity, balance, controllability, observability and robustness are actually associated with long-range dependence phenomena. related development has been made in different fields in this article. The scope of the booklet is hence to provide the state-of-the-art within the research of fractional platforms and the applying of fractional differentiation.

As this quantity covers fresh functions of fractional calculus, it will likely be of curiosity to engineers, scientists, and utilized mathematicians.

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**Extra resources for Advances in fractional calculus**

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Rendiconti Academia Nazionale dei Lincei Series V, Vol. 13, pp. 3–5. 3. Humbert P (1953) Quelques resultants relatifs a la fonction de Mittag-Leffler. Comptes Rendus de l’Academie des Sciences, Paris, Vol. 236, pp. 1467–1468. 4. Agarwal RP (1953) A propos d’une note de M. Pierre Humbert Comptes Rendus de l’Academie des Sciences, Paris, Vol. 236, pp. 2031–2032. 5. Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG (1955) Higher Transcendental Functions, Vol. 3. McGraw-Hill, New York, pp. 206–212. 6.

Equations (14) and (15) are the main results of this paper. 5 Accuracy of the Iteration Results Using Eqs. (14) and (15), the number of real zeros of E ,1 (–x) can be calculated for arbitrary in the range 1 < < 2 with some restrictions based on the number of significant digits in . These restrictions result because of the approximate solution of E , (–x) = 0 used in the derivation. The approximation that f , (–x) 0 in Eq. (11) improves as approaches 2 and consequently the results of using Eqs. (14) and (15) become more accurate.

5a), of E ,1(–x) as a sum of two functions g(–x) and f(–x). The function f ,1(–x) is negative for all x and is a completely monotonic function which decreases toward zero with increasing x [14]. The function g ,1(–x) exhibits oscillations with an amplitude which decays exponentially. Each full period oscillation of the cos[x 1/ sin( / )] term in g ,1(–x) results in two zeros for when cosine is positive and g ,1(–x) is larger than f ,1(–x) it gives rise to a relative maximum in E ,1(–x) above the x-axis.