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【文章內(nèi)容簡介】 ative by first looking at examples of familiar nth order derivatives like and then replacing the natural number n by other numbers like In this way, like detectives, we will try to see what mathematical structure might be hidden in the idea. We will avoid a formal definition of the fractional derivative until we have first explored the possibility of various approaches to the notion. (For a quick look at formal definitions see the excellent expository paper by Miller [8].) As the exploration continues, we will at times ask the reader to ponder certain questions. The answers to these questions are found in the last section of this paper. So just what is a fractional derivative? Let us see. . . . Fractional derivatives of exponential functions We will begin by examining the derivatives of the exponential function because the patterns they develop lend themselves to easy exploration. We are familiar with the expressions for the derivatives of ., and, in general, when n is an integer. Could we replace n by 1/2 and write Why not try? Why not go further and let n be an irrational number like or a plex number like1+i ? We will be bold and write (1) for any value of , integer, rational, irrational, or plex. It is interesting to consider the meaning of (1) when is a negative integer. We naturally want .Since ,we have .Similarly, ,so is it reasonable to interpret when is a negative integer –n as the nth iterated integral. represents a derivative if is a positive real number and an integral if is a negative real number. Notice that we have not yet given a definition for a fractional derivative of a general function. But if that definition is found, we would expect our relation (1) to follow from it for the exponential function. We note that Liouville used this approach to fractional differentiation in his papers [5] and [6]. Questions Q1 In this case does ? Q2 In this case does ? Q3 Is , and is ,(as listed above) really true, or is there something missing? Q4 What general class of functions could be differentiated fractionally be means of the idea contained in (1)? Trigonometric functions: sine and cosine. We are familiar with the derivatives of the sine function: This presents no obvious pattern from which to find . However, graphing the functions discloses a pattern. Each time we differentiate, the graph of sin x is shifted to the left. Thus differentiating sin x n times results in the graph of sin x being shifted to the left and so . As before, we will replace the positive integer n with an arbitrary . So, we now have an expression for the general derivative of the sine function, and we can deal similarly with the cosine: (2) After finding (2), it is natural to ask if these guesses are consistent with the results of the previous section for the exponential. For this purpose we can use Euler’s expression, Using (1) we can calculate which agrees with (2). Question Q5 What is ? Derivatives of We now look at derivatives of powers of x. Starting with we have: Multiplying the numerator and denominator of (3) by (pn)! results in This is a general expression of .To replace the positive integer n by the arbitrary number we may use the gamma function. The gamma function gives meaning to p! and (pn)! in (4) when p and n are not natural numbers. The gamma function was introduced by Euler in the 18th century to generalize the notion of z! to noninteger values of z. Its definition is ,and it has the property that . We can rewrite (4) as which makes sense if n is not an integer, so we put for any . With (5) we can extend the idea of a fractional derivative to a large number of functions. Given any function that can be expanded in a Taylor series in powers of x, assuming we c
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