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【正文】 colePolytech., 13 (1832), 71162.7. A. C. McBride and G. F. Roach, Fractional Calculus, Pitman Publishing, 1985.8. K. S. Miller, Derivatives of noninteger order, Math. Mag., 68 (1995), 183192.9. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and FractionalDifferential Equations, John Wiley amp。ometrie et de m233。1A CHILD’S GARDEN OF FRACTIONAL DERIVATIVESThe College Mathematics Journal, Vol. 31, No. 2, (2020), pp. 82887Marcia Kleinz and Thomas J. Osler1. Introduction. We are all familiar with the idea of derivatives. The usual notation df xdx() or Df x1( ) , dfxdx22()or Dfx2()is easily understood. We are also familiar with properties like Df x f y Df x Df yb g b g b g b g+=+ .But what would be the meaning of notation like dfxdx1212//() or Dfx12/() ?Most readers will not have encountered a derivative of “order 1/2” before, becausealmost none of the familiar text books mention it. Yet the notion was discussed briefly asearly as the eighteenth century by Leibniz. Other giants of the past including L’Hospital,Euler, Lagrange, Laplace, Riemann, Fourier, Liouville, and others at least toyed with theidea. Today a vast literature exists on this subject called the “fractional calculus”. Twotext books on the subject at the graduate level have appeared recently , [9] and [11]. Alsotwo collections of papers delivered at conferences are found in [7] and [14]. A set ofvery readable seminar notes has been prepared by Wheeler [15], but these have not beenpublished.It is the purpose of this paper to introduce the fractional calculus in a gentle manner.Rather than the usual definition lemma theorem approach, we explore the idea of afractional derivative by first looking at examples of familiar nth order derivatives2like De aenax nax= and then replacing the natural number n by other numbers like 1/2.In this way , like detectives, we will try to see what mathematical structure might behidden in the idea. We will avoid a formal definition of the fractional derivative until wehave first explored the possibility of various approaches to the notion. (For a quick lookat formal definitions see the excellent expository paper by Miller [8].)As the exploration continues, we will at times ask the reader to ponder certainquestions. The answers to these questions are found in the last section of this paper. So just what is a fractional derivative? Let us see... .2. Fractional derivatives of exponential functions. We will begin by examining thederivatives of the exponential function eax because the patterns they develop lendthemselves to easy exploration. We are familiar with the expressions for the derivativesof eax. De aeax ax1= , De aeax ax22= , De aeax ax33= , and in general, De aenax nax= whenn is an integer. Could we replace n by 1/2 and write De aeax ax12 12//= ? Why not try?Why not go further and let n be an irrational number like 2 , or a plex number like1+i ? We will be bold and write (1) De aeax axαα=for any value of α , integer, rational, irrational, or plex. It is interesting to considerthe meaning of (1) when α is
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