外文翻译-分数阶导数[经典]

外文翻译-分数阶导数[经典] 译文: 分数阶导数的儿童乐园 Marcia Kleinz , Thomas J. Osler 大学 数学 数学高考答题卡模板高考数学答题卡模板三年级数学混合运算测试卷数学作业设计案例新人教版八年级上数学教学计划 学报(美国)，2000年3月，31卷，第2期，第82-88 页 1引言 2dfx()dfx()12或Dfx()或Dfx()2dxdx我们都熟悉的导数的定义。通常记作 这些都是很容 DfxfyDfxDfy[()()]()()，,，易理解的。我们同样也熟悉一些有关导数的性质，例如但是 1/2dfx()1/2或者D()fx像这样的记号又代表什么意思呢,大多数的读者之前肯定没有遇到过导1/2dx 数的阶数是1/2的。因为几乎没有任何教科书会提到它。然而，这个概念早在18世纪，Leibnitz已经开始探讨。在之后的岁月里，包括L’Hospital, Euler,Lagrange, Laplace, Riemann, Fourier, Liouville等数学大家和其他一些数学家也出现过或者研究过的概念。现在，关于―分数微积分‖的文献已经大量存在。近期关于―分数微积分‖的两本研究生教材也出版了，就是参考文献[9]和[11]。此外，两篇在会议上发表的论文[7]和[14]也被收录。Wheeler在文献[15]已编制了一些可读性较强，较易理解的资料，虽然这些都还没有正式出版。 本论文的目的是想用一种亲和的口吻去介绍分数阶微积分。而不是像平常教科书里面的从定义-引理-定理的方法介绍它。我们寻找了一个新的想法去介绍分数阶导数。首先我们从熟悉 naxnaxDeae,的n阶导数的例子开始，比如。然后用其他数字取代自然数字n。这种方式，感觉像是侦探一样，步步深入。我们将寻求蕴含在这个构思里面的数学结构。我们在探讨了各种思路，对分数阶导数的概念后，才对分数阶导数给出正式定义。(如果想快速浏览它的正式定义，请参见米勒的优秀论文，参考文献[8]。) 随着探究的深入，我们会不时地让读者去思考一些问题。对这些问题的答案将在本文的最后一节呈现。那到底什么是一个分数阶导数呢,让我们一起来看看吧…… 2指数函数的分数阶导数 axaxee我们将首先研究指数函数的导数。因为他们导数的形式，比较容易推广。我们熟悉 12233axaxaxaxaxaxDeaeDeaeDeae,,,,,的导数的表达式。，在一般情况下，当n为整数时， naxnax1/21/2axaxDeae,Deae,。那么我们能不能用1/2取代n，并记作呢,我们何不尝试一下,为什么不更进一步，让n是一个无理数或者复数比如1+i, 我们大胆地写作 ,,axaxDeae,， (1) ,,对任意一个，无论是整数，有理数，无理数，还是复数。当是负整数时，考虑(1) 1axaxaxax,1eDe,(())eDDe,(())式的意义是很有趣的。我们自然希望有成立。因为，所以a ,1axax,2axaxDeedx(),Deedxdx(),,,,,,D我们有。同理。当是负整数时，我们将看作是 ,,,,DDn次迭代的积分是合理。当是正实数，代表导数，当是负实数，代表积分。 请注意，我们还没对一般函数给出分数阶导数的定义。但是，如果这一定义被发现，我们期望指数函数的分数阶导数遵循关系式(1)。我们注意到，刘维尔在他的论文[5]和[6]中就是采用这种方法去考虑微分的。 问题 axaxaxax,1212DcececDecDe()，,，1212问题1:在上述情况下，成立吗, ,,,,axax，DDeDe,问题2:在上述情况下，成立吗, ,1axax,2axaxDeedx(),Deedxdx(),,,,问题3:上述和，真的正确吗,还是遗漏了一些东西, 问题4:用蕴含在(1)式的想法，怎样对一般性的函数求分数阶导数, 3三角函数:正弦函数和余弦函数 012DxxDxxDxxsinsin,sincos,sinsin,,,,,？我们对于正弦函数的导数很熟悉: 1/2Dxsin这些对于寻求，并没有明显的规律。但是，当我们画出这些函数的图形时，会挖掘出 sinx,/2sinx其中的规律。即每当我们求一次微分，的图像向左平移。所以对求n次微分， n,nsinsin(),，Dxxsinxn,/22那么得到的图像就是向左平移，即得到。如前，我们用任 ,,意数替换正整数n。所以，我们得到正弦函数的任意次导数的表达式，同理我们也得到余弦函数的: ,,,,,,DxxDxxsinsin(),coscos().,，,，22 (2) 在得到表达式(2)之后，我们自然想，这个猜测与指数函数的结果是否保持一致。为了 ixexix,，cossin验证这个猜测，我们可以使用欧拉公式。利用表达式(1)，我们可以计算得 ,,,,,,,,ixixiix(/2)Deieeexix,,,，，，cos()sin()22到，这与(2)式是吻合的。 问题 ,Daxsin()问题5:是什么, px4的导数 px我们现在看看x次方的导数。我们以为例有: 012ppppppDxxDxpxDxppx,,,,,,(1),,？ nppn,Dxppppnx,,,,，(1)(2)(1).(3)？ 表达式(3)用连乘的分子和分母去替换，则得到结果如下 ()!pn, ppppnpnpnp(1)(2)(1)()(1)1!,,,，,,,？？ppnpn,, xxx,,(4)()(1)1()!pnpnpn,,,,？ np,Dx上式就是的一般表达式。我们通过伽玛函数，用任意数替换正整数n。当(4)式中的p和n是不是自然数时，伽玛函数使他们在替换后任然有意义。伽马函数是欧拉在18世纪引进 ,tz1,,z!的概念。当时是推广记号，当z不是整数时。它的定义是，它具有这样的,,()dzett,0 性质。 ,,(+1)!zz ,，(1)pnppn,那么我们可以将表达式(4)重新写作这使得当n不是整数式，Dxx,,,,，(1)pn ,(4)式还是有意义的。所以对于任意的，我们写作 ,，(1)p,,pp, Dxx,(5),,,，(1)p 利用(5)式，我们可以将分数阶导数延伸到很多的函数。因为对于任意给定的函数，我 ,nfx()们可以利用Taylor级数展开成多项式的形式，假设我们可以对进行任意fxax(),,,n0n, 次微分，那么我们得到 ,,,，(1)n,,,nn, DfxaDxax().(6),,,,nn,,,，(1)n00nn,, 最终那个表达式(6)呈现出具有作为分数阶导数定义候选项的气质。因为大量的函数都可以利用Taylor公式展开成幂级数的形式。然后，我们很快会发现它会导致矛盾的产生。 问题 ,问题6:是否有几何意义, Dfx() 5一个神秘的矛盾 x我们将的分数阶导数写为 e ,xx Dee,(7) 现在让我们拿它与(6)式进行对比，看看他们是否一致。 ,1xn从Taylor级数来看，结合(6)式，我们得到如下表达式 ex,,,n!0n, n,x,x ,.(8)De,,,,，(1)n0n, xe但是，(7)及(8)是不等价的，除非是整数。当是整数时，(8)式的右侧是的级,, 数形式，只是用不同的表达方式。但是当不是整数时，我们得到两个完全不一样的函数。我, 们发现了历史上引起大问题的矛盾。这看起来好像我们，指数函数的分数阶导数的表达式(1)与次方函数的分数阶导数的公式(6)是相互矛盾。 正是因为有这样一个矛盾，所以分数阶微积分一般不会出现在初等阶段的教科书里面。在传统的微积分中，导数的次数是整数次的，求导的函数是初等函数。不幸的是，在分数阶微积分中，这是不正确的。通常，一个初等函数的分数阶导数是较高级的超越函数。关于分数阶导数的表格，请参阅文献[3]。 此时，您可能会问我们怎么继续探究呢,这个谜团将在之后的部分中被解决。敬请关注…… 6多重迭代积分 ,1Dfxfxdx()(),我们一直在谈论导数。积分也是反复被提及的。我们可以写，但是等, x,1Dfxftdt()(),式右边是不确定的。我们可以写作。第二次积分可以写成,0 xt2,2Dfxftdtdt()(),。积分区域是图1中的三角形。如果我们交换积分的顺序，那么112,,00 xx,2图1的右侧图可以表现出。 Dfxftdtdt()(),121,,0t1 因为不是一个关于的函数，所以可以将里面的积分移到外面，即 tft()21 xxx,2 Dfxftdtdtftxtdt()()()(),,,121111,,,00t1 x,2或者Dfxftxtdt()()(),,。 ,0 使用相同的过程步骤，我们可以写出 xx11,,3243DfxftxtdtDfxftxtdt,,,, ()()(),()()(),,,00,223 在一般情况下， x1,,1nn Dfxftxtdt,,()()().,0n,(1)! 现在，我们用先前做的方法，用任意数替换，用伽玛函数替换阶乘，然后得到,,n x1()ftdt, ,Dfx().(9)，1,,0,,,,()()xt 这个一般性的表达式(使用积分)的分数阶导数表达式，有成为定义的潜力。但是存在一 ,,0个问题。如果该积分是反常积分。因为当对任意，积分是发散,,,1,txxt,,,,0.的。当反常积分收敛。所以当,是负数时，原表达式是正确的。因此当,是负数,,,10,, 时(9)式收敛，即它是一个分数阶次积分。 在我们结束这一部分之前，需要提下，趋于零的下极限是任意的。可以简单的认为存在下b极限。但是会造成最后结果表达式的不同。正因为如此，很多这个领域的研究人员使用符号,bxDfx()。这个符号说明了极限过程是从到的。这样我们从(9)式得到 bx x1()ftdt,, Dfx().(10)bx，1,,b,,,,()()xt 问题 b问题7:如下分数阶微分的下极限是什么, ,，(1)p,,pp, Dxcxc()(),,,bx,,,，(1)p 7解秘 现在，你可以开始去发现前面哪些地方出错了。我们对于分数阶积分包含极限，并不感到惊讶。因为积分是涉及到极限的。然而普通的导数不涉及积分的极限，没有人希望分数阶导数 ,包含这样的极限。我们认为，导数是函数的局部性质。分数阶导数的符号既包含导数(是D,正数)又有积分(是负数)。积分是处于极限之中的。事实证明，分数阶导数也是处于极限, 之中的。出现该对矛盾的原因是，我们使用了两种不同的极限。 现在，我们可以解决这个谜团了。 秘密是什么,让我们停下来想一想。表达式(1)中指数函数起作用的极限是什么,记得我们要期望写成 x1,1axaxaxDeedxe,, .(11)bx,ba x11axaxabedxee,,b取什么值时，将得到这个答案,由于在(11)式中积分就是 .,baa 1abab,,,.b,,,0e,为了得到我们想要的形式，只有当时。即如果a是正数，那么。a 这种类型的拥有下极限为的积分，有时也称为Weyl分数阶导数。从(10)的符号，我们可,, ,,axax以将(1)写做 Deae,.,,x pp，，11xxbp,1ppx极限在公式(5)的导数中是起什么作用,我们有 ,,,.Dxxdxbx,b，，11pp p，1bb,0同样，我们希望。当时，结论是成立的。所以我们觉得将(5)的符号写成,0p，1 p,,,，(1)px,pp,Dx更准确。因此，表达式(5)中隐含了的下极限为0。然而，表Dx,0x,,,，(1)p ,axDe达式(1)中的下极限为。这个差异就是(7)和(8)为什么不等价的原因。在(7)-, ax,ax,中我们计算，在(8)中我们计算。 DeDe,,0xx 如果读者希望继续这一研究，我们推荐Miller的一篇很好的论文[8]，和由Oldham和Spanie合著的优秀图书[11]，以及Miller和Ross合著的优秀图书[9]。这两本书都包含了从很多文献角度分数阶微积分简短精要的历史。由Miller和Ross合著的图书[9]很好地讨论了分数阶微分方程。Wheeler的注记[14]是另一个这方面一流的介绍性文章，具有很高的推广普及价值。Wheeler给出了几个简单的应用例子，而且阅读起来非常有趣。其他的历史性文献请参考[1，2，4，5，6，10，13]。 8问题的解答 以下是文章中8个问题的简短回答。 问题1:成立的，这个性质是保持的。 问题2:成立的，这个由关系式(2.2)很容易证明。 问题3:遗漏了一些东西。遗漏的是积分常数。应该是这样的 ,,,,1122axaxaxaxaxax DeedxaecDeedxaecxc,,，,,，，？112,,, ,inx问题4:将展开成傅立叶级数形式，。假设我们可以连续次地取分fx()fxce(),,nn,,, ,,inx数阶微分，我们得到。 Dfxcine()(),,,nn,,, ,,问题5: 。 Daxaaxsin()sin(/2),，,, 12问题6:我们知道表示的曲线斜率，表示曲线的凹凸性。但三yfx,()Dfx()Dfx() ,阶和更高阶导数给出的几何意义很少或根本没有。那么对于这些特殊的导数，分数阶Dfx()导数没有简单的几何意义对我们来说也并不感到惊讶了。 ""c问题7:分数阶微分的下极限是。 【参考文献】 [1] A. K. Grunwald, Uber begrenzte Derivationen und deren Anwendung, Z. Angew.Math. Phys., 12 (1867), 441–480. [2]O. Heaviside, Electromagnetic Theory, vol. 2, Dover, 1950, Chap. 7, 8. [3]J. L. Lavoie, T. J. Osler and R. Trembley, Fractional derivatives and special functions, SIAM Rev., 18 (1976), 240–268. [4]A. V. Letnikov, Theory of differentiation of fractional order, Mat. Sb., 3 (1868),1–68. [5]J. Liouville, Memoire sur quelques questions de géometrie et de mécanique, et su run noveau gentre pour resoudre ces questions, J. École Polytech., 13(1832), 1–69. [6]J. Liouville, Memoire: sur le calcul des differentielles á indices quelconques, J.École Polytech., 13(1832), 71–162. [7]A. C. McBride and G. F. Roach, Fractional Calculus, Pitman, 1985. [8]K. S. Miller Derivatives of noninteger order, Math. Mag., 68 (1995), 183–192. [9]K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, 1993. [10]P. A. Nekrassov, General differentiation, Mat. Sb., 14 (1888), 45–168. [11]K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, 1974. [12]T. J. Osler, Fractional derivatives and the Leibniz rule, Amer. Math. Monthly, 78 (1971), 645–649. [13]E. L. Post, Generalized differentiation, Trans. Amer. Math. Soc., 32 (1930) 723–781. [14]B. Ross, editor, Proceedings of the International Conference on Fractional Calculusand its applications, Springer-Verlag, 1975. [15]N. Wheeler, Construction and Physical Application of the fractional Calculus, notes for a Reed College Physics Seminar, 1997. 浙江师范大学本科毕业设计(论文)外文翻译 原文: A Child’s Garden of Fractional Derivatives Marcia Kleinz and Thomas J. Osler The College Mathematics Journal, March 2000, Volume 31, Number 2, pp. 82–88 Marcia Kleinz is an instructor of mathematics at Rowan University. Marcia is married and has two children aged four and eight. She would rather research the fractional calculus than clean, and preparing lectures is preferable to doing laundry. Her hobbies include reading, music, and physical fitness. Tom Osler (osler@rowan.edu) is a professor of mathematics at Rowan University. He received his Ph.D. from the Courant Institute at New York University in 1970 and is the author of twenty-three mathematical papers. In addition to teaching university mathematics for the past thirty-eight years, Tom has a passion for long distance running. Included in his over 1600 races are wins in three national championships in the late sixties at distances from 25 kilometers to 50 miles. He is the author of two running books. Introduction We are all familiar with the idea of derivatives. The usual notation 2dfx()dfx()12or ， or Dfx()Dfx()2dxdx is easily understood. We are also familiar with properties like DfxfyDfxDfy[()()]()()，,， 1/2dfx()1/2But what would be the meaning of notation like or ,Most readers will not Dfx()1/2dx have encountered a derivative of ―order ‖ before, because almost none of the familiar textbooks mention it. Yet the notion was discussed briefly as early as the eighteenth century by Leibnitz. Other giants of the past including L’Hospital, Euler, Lagrange, Laplace, Riemann, Fourier, Liouville, and others at least toyed with the idea. Today a vast literature exists on this subject called the ―fractional calculus.‖ Two text books on the subject at the graduate level have appeared recently, [9] and [11]. Also, two collections of papers delivered at conferences are found in [7] and [14]. A set of very 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 a fractional derivative by first naxnaxDeae,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 axeWe 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 12233axaxaxaxaxaxaxnaxnaxDeaeDeaeDeae,,,,,eDeae,of ., and, in general, when n is an 1/21/2axaxDeae,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 complex number like1+i ? 2 We will be bold and write ,,axaxDeae,， (1) ,for any value of , integer, rational, irrational, or complex. It is interesting to consider the meaning axax,1eDDe,(()),of (1) when is a negative integer. We naturally want .Since 1axax,1axax,2axaxeDe,(())Deedx(),Deedxdx(),,,,we have .Similarly, ,so is it a, ,,Dreasonable to interpret when is a negative integer –n as the nth iterated integral. ,,,Drepresents 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 axaxaxax,1212DcececDecDe()，,，1212Q1 In this case does ? ,,,,axax，DDeDe,Q2 In this case does ? ,1axax,2axaxDeedx(),Deedxdx(),,,,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: 012DxxDxxDxxsinsin,sincos,sinsin,,,,,？ 1/2DxsinThis presents no obvious pattern from which to find . However, graphing the functions ,/2discloses a pattern. Each time we differentiate, the graph of sin x is shifted to the left. Thus n,/2differentiating sin x n times results in the graph of sin x being shifted to the left and so n,nsinsin(),，Dxx,2. 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: ,,,,,,DxxDxxsinsin(),coscos().,，,，22 (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, ixexix,，cossin Using (1) we can calculate ,,,,,,,,ixixiix(/2)Deieeexix,,,，，，cos()sin()22 which agrees with (2). Question ,Daxsin()Q5 What is ? pxDerivatives of pxWe now look at derivatives of powers of x. Starting with we have: 012ppppppDxxDxpxDxppx,,,,,,(1),,？ nppn,Dxppppnx,,,,，(1)(2)(1).(3)？ Multiplying the numerator and denominator of (3) by (p-n)! results in ppppnpnpnp(1)(2)(1)()(1)1!,,,，,,,？？ppnpn,, xxx,,(4)()(1)1()!pnpnpn,,,,？ np,This is a general expression of .To replace the positive integer n by the arbitrary number Dx we may use the gamma function. The gamma function gives meaning to p! and (p-n)! in (4) when p and n are not natural numbers. The gamma function was introduced by Euler in the 18th century to ,tz1,,generalize the notion of z! to non-integer values of z. Its definition is ,and it has ,,()dzett,0 the property that . ,,(+1)!zz We can rewrite (4) as ,，(1)pnppn, Dxx,,,,，(1)pn which makes sense if n is not an integer, so we put ,，(1)p,,pp, Dxx,(5),,,，(1)p ,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, ,n fxax(),,,n0n, assuming we can differentiate term by term we get ,,,，(1)n,,,nn, DfxaDxax().(6),,,,nn,,,，(1)n00nn,, The final expression presents itself as a possible candidate for the definition of the fractional derivative for the wide variety of functions that can be expanded in a Taylor’s series in powers of x. However, we will soon see that it leads to contradictions. Question ,Q6 Is there a meaning for in geometric terms? Dfx() A mysterious contradiction We wrote the fractional derivative of as ,xx Dee,(7) ,1xnLet us now compare this with (6) to see if they agree. From the Taylor Series, (6) ex,,,n!0n,gives n,x,x ,.(8)De,,,,，(1)n0n, But (7) and (8) do not match unless is a whole number! When is a whole number, the right side of (8) xwill be the series of with different indexing. But when is not a whole number, we have two e, entirely different functions. We have discovered acontradiction that historically has caused great problems. It appears as though ourexpression (1) for the fractional derivative of the exponential is inconsistent with ourformula (6) for the fractional derivative of a power. This inconsistency is one reason the fractional calculus is not found in elementary texts. In the traditional calculus, where is a whole number, the derivative of an elementary function is an , elementary function. Unfortunately, in the fractional calculus this is not true. The fractional derivative of an elementary function is usually a higher transcendental function. For a table of fractional derivatives see [3]. At this point you may be asking what is going on? The mystery will be solved in later sections. Stay tuned . . . . Iterated integrals We have been talking about repeated derivatives. Integrals can also be repeated. We could write ,1,but the right-hand side is indefinite. We will instead write Dfxfxdx()(),, xxt2,1,2Dfxftdt()(),Dfxftdtdt()(),.The second integral will then be .112,,,000 The region of integration is the triangle in Figure 1. If we interchange the order of integration, the right-hand diagram in Figure 1 shows that xx,2 Dfxftdtdt()(),121,,0t1 tft()Since is not a function of , it can be moved outside the inner integral so,21 xxx,2 Dfxftdtdtftxtdt()()()(),,,121111,,,00t1 or x,2 Dfxftxtdt()()(),,,0 Using the same procedure we can show that xx11,,3243 DfxftxtdtDfxftxtdt,,,,()()(),()()(),,,00,223 and, in general, x1,,1nn Dfxftxtdt,,()()().,0n,(1)! Now, as we have previously done, let us replace the –n with arbitraryand the factorial with the , gamma function to get x1()ftdt, ,Dfx().(9)，1,,0,,,,()()xt This is a general expression (using an integral) for fractional derivatives that has the potential of being used as a definition. But there is a problem. If the integer is improper. This occurs because ,,,1, ,,0as The integral diverges for every .When the improper txxt,,,,0.,,,10,, integral converges, so if is negative there is no problem. Since (9) converges only for negative it , is truly a fractional integral. Before we leave this section we want to mention that the choice of zero for the lower limit was arbitrary. The lower limit could just as easily have been b. However, the resulting expression will be different. Because of this, many people who work in this field use the ,notation indicating limits of integration going from b to x. Thus we have from (9)Dfx()bx x1()ftdt, ,Dfx().(10)bx，1,,b,,,,()()xt Question Q7 What lower limit of fractional differentiation b will give us the result ,，(1)p,,pp,? Dxcxc()(),,,bx,,,，(1)p The mystery solved Now you may begin to see what went wrong before. We are not surprised that fractional integrals involve limits, because integrals involve limits. Since ordinary derivatives do not involve limits of integration, no one expects fractional derivatives to involve such limits. We think of derivatives as ,Dlocal properties of functions. The fractional derivative symbolincorporates both derivatives ,,(positive) and integrals (negative). Integrals are between limits. It turns out that fractional derivatives are between limits also. The reason for the contradiction is that two different limits of integration were being used. Now we can resolve the mystery. What is the secret? Let’s stop and think. What are the limits that will work for the exponential from (1)? Remember we want to write x1,1axaxax Deedxe,,.(11)bx,ba What value of b will give this answer? Since the integral in (11) is really x11axaxab edxee,,.,baa 1abab,,,.we will get the form we want when 0.It will be zero when So, if a is positive, e,a b,,,then.This type of integral with a lower limit of is sometimes called the Weyl fractional ,, derivative. In the notation from (10) we can write (1) as ,,axax Deae,.,,x pNow, what limits will work for the derivative of in (5)? We have x pp，，11xxb,1pp ,,,.Dxxdxbx,b，，11pp p，1bb,0Again we want 。This will be the case when. We conclude that (5) should be written ,0p，1 in the more revealing notation p,,,，(1)pxp, Dx,0x,,,，(1)p ,pDxSo, the expression (5) for has a built-in lower limit of 0. However, expression (1) for ,axDehasas a lower limit. This discrepancy is why (7) and (8) do not match. In-, ax,ax,(7) we calculated and in (8) we calculated. DeDe,,0xx If the reader wishes to continue this study, we recommend the very fine paper by Miller [8] as well as the excellent books by Oldham and Spanier [11] and by Miller and Ross [9]. Both books contain a short, but very good history of the fractional calculus with many references. the book by Miller and Ross [9] has an excellent discussion of fractional differential equations. Wheeler’s notes [14] are another first rate introduction, which should be made more widely available. Wheeler gives several easily accessible applications, and is particularly interesting to read. Other references of historical interest are [1, 2, 4, 5, 6, 10, 13]. Answers to questions The following are short answers to the questions throughout the paper. Q1 Yes, this property does hold. Q2 Yes, and this is easy to show from relation (2.2). Q3 Something is missing. That something is the constant of integration. We should have ,,,112axaxaxaxDeedxaecDe,,，1, axax,2？,,，，edxaecxc12,, ,inxQ4 Let be expandable in an exponential Fourier series, . Assuming we can fxce(),,nn,,, ,,inxdifferentiate fractionally term by term we get Dfxcine()(),,,nn,,, ,,Q5 Daxaaxsin()sin(/2),，,, 1Q6 We know thatis geometrically interpreted as the slope of the curveand yfx,()Dfx() 2gives us the concavity of the curve. But the third and higher derivatives give us little or no Dfx() ,geometric information. Since these are special cases of we are not surprised that there is Dfx() no easy geometric meaning for the fractional derivative. Q7 The lower limit of differentiation should be ―c‖. References 1. A. K. Grunwald, Uber begrenzte Derivationen und deren Anwendung, Z. Angew. Math. Phys., 12 (1867), 441–480. 2. O. Heaviside, Electromagnetic Theory, vol. 2, Dover, 1950, Chap. 7, 8. 3. J. L. Lavoie, T. J. Osler and R. Trembley, Fractional derivatives and special functions, SIAM Rev., 18 (1976), 240–268. 4. A. V. Letnikov, Theory of differentiation of fractional order, Mat. Sb., 3 (1868), 1–68. 5. J. Liouville, Memoire sur quelques questions de géometrie et de mécanique, et surun noveau gentre pour resoudre ces questions, J. École Polytech., 13(1832), 1–69. 6. J. Liouville, Memoire: sur le calcul des differentielles á indices quelconques, J.École Polytech., 13(1832), 71–162. 7. A. C. McBride and G. F. Roach, Fractional Calculus, Pitman, 1985. 8. K. S. Miller Derivatives of noninteger order, Math. Mag., 68 (1995), 183–192. 9. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, 1993. 10. P. A. Nekrassov, General differentiation, Mat. Sb., 14 (1888), 45–168. 11. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, 1974. 12. T. J. Osler, Fractional derivatives and the Leibniz rule, Amer. Math. Monthly, 78 (1971), 645–649. 13. E. L. Post, Generalized differentiation, Trans. Amer. Math. Soc., 32 (1930) 723–781. 14. B. Ross, editor, Proceedings of the International Conference on Fractional Calculus and its applications, Springer-Verlag, 1975. 15. N. Wheeler, Construction and Physical Application of the fractional Calculus, notes for a Reed College Physics Seminar, 1997.