數(shù)學(xué)專業(yè)外文翻譯----分?jǐn)?shù)階導(dǎo)數(shù)的兒童樂(lè)園.doc

浙江師范大學(xué)本科畢業(yè)設(shè)計(jì)(論文)外文翻譯譯文：分?jǐn)?shù)階導(dǎo)數(shù)的兒童樂(lè)園Marcia Kleinz , Thomas J. Osler大學(xué)數(shù)學(xué)學(xué)報(bào)（美國(guó)），2000年3月，31卷，第2期，第82-88 頁(yè)1引言我們都熟悉的導(dǎo)數(shù)的定義。通常記作 這些都是很容易理解的。我們同樣也熟悉一些有關(guān)導(dǎo)數(shù)的性質(zhì)，例如但是像這樣的記號(hào)又代表什么意思呢？大多數(shù)的讀者之前肯定沒(méi)有遇到過(guò)導(dǎo)數(shù)的階數(shù)是1/2的。因?yàn)閹缀鯖](méi)有任何教科書會(huì)提到它。然而，這個(gè)概念早在18世紀(jì)，Leibnitz已經(jīng)開(kāi)始探討。在之后的歲月里，包括LHospital, Euler,Lagrange, Laplace, Riemann, Fourier, Liouville等數(shù)學(xué)大家和其他一些數(shù)學(xué)家也出現(xiàn)過(guò)或者研究過(guò)的概念?，F(xiàn)在，關(guān)于“分?jǐn)?shù)微積分”的文獻(xiàn)已經(jīng)大量存在。近期關(guān)于“分?jǐn)?shù)微積分”的兩本研究生教材也出版了，就是參考文獻(xiàn)9和11。此外，兩篇在會(huì)議上發(fā)表的論文7和14也被收錄。Wheeler在文獻(xiàn)15已編制了一些可讀性較強(qiáng)，較易理解的資料，雖然這些都還沒(méi)有正式出版。本論文的目的是想用一種親和的口吻去介紹分?jǐn)?shù)階微積分。而不是像平常教科書里面的從定義-引理-定理的方法介紹它。我們尋找了一個(gè)新的想法去介紹分?jǐn)?shù)階導(dǎo)數(shù)。首先我們從熟悉的n階導(dǎo)數(shù)的例子開(kāi)始，比如。然后用其他數(shù)字取代自然數(shù)字n。這種方式，感覺(jué)像是偵探一樣，步步深入。我們將尋求蘊(yùn)含在這個(gè)構(gòu)思里面的數(shù)學(xué)結(jié)構(gòu)。我們?cè)谔接懥烁鞣N思路，對(duì)分?jǐn)?shù)階導(dǎo)數(shù)的概念后，才對(duì)分?jǐn)?shù)階導(dǎo)數(shù)給出正式定義。（如果想快速瀏覽它的正式定義，請(qǐng)參見(jiàn)米勒的優(yōu)秀論文，參考文獻(xiàn)8。）隨著探究的深入，我們會(huì)不時(shí)地讓讀者去思考一些問(wèn)題。對(duì)這些問(wèn)題的答案將在本文的最后一節(jié)呈現(xiàn)。那到底什么是一個(gè)分?jǐn)?shù)階導(dǎo)數(shù)呢？讓我們一起來(lái)看看吧2指數(shù)函數(shù)的分?jǐn)?shù)階導(dǎo)數(shù)我們將首先研究指數(shù)函數(shù)的導(dǎo)數(shù)。因?yàn)樗麄儗?dǎo)數(shù)的形式，比較容易推廣。我們熟悉的導(dǎo)數(shù)的表達(dá)式。，在一般情況下，當(dāng)n為整數(shù)時(shí)，。那么我們能不能用1/2取代n，并記作呢？我們何不嘗試一下？為什么不更進(jìn)一步，讓n是一個(gè)無(wú)理數(shù)或者復(fù)數(shù)比如1+i？我們大膽地寫作 （1）對(duì)任意一個(gè)，無(wú)論是整數(shù)，有理數(shù)，無(wú)理數(shù)，還是復(fù)數(shù)。當(dāng)是負(fù)整數(shù)時(shí)，考慮（1）式的意義是很有趣的。我們自然希望有成立。因?yàn)椋晕覀冇?。同理。?dāng)是負(fù)整數(shù)時(shí)，我們將看作是n次迭代的積分是合理。當(dāng)是正實(shí)數(shù)，代表導(dǎo)數(shù)，當(dāng)是負(fù)實(shí)數(shù)，代表積分。請(qǐng)注意，我們還沒(méi)對(duì)一般函數(shù)給出分?jǐn)?shù)階導(dǎo)數(shù)的定義。但是，如果這一定義被發(fā)現(xiàn)，我們期望指數(shù)函數(shù)的分?jǐn)?shù)階導(dǎo)數(shù)遵循關(guān)系式（1）。我們注意到，劉維爾在他的論文5和6中就是采用這種方法去考慮微分的。問(wèn)題問(wèn)題1：在上述情況下，成立嗎？問(wèn)題2：在上述情況下，成立嗎？問(wèn)題3：上述和，真的正確嗎？還是遺漏了一些東西？問(wèn)題4：用蘊(yùn)含在（1）式的想法，怎樣對(duì)一般性的函數(shù)求分?jǐn)?shù)階導(dǎo)數(shù)？3三角函數(shù)：正弦函數(shù)和余弦函數(shù)我們對(duì)于正弦函數(shù)的導(dǎo)數(shù)很熟悉：這些對(duì)于尋求，并沒(méi)有明顯的規(guī)律。但是，當(dāng)我們畫出這些函數(shù)的圖形時(shí)，會(huì)挖掘出其中的規(guī)律。即每當(dāng)我們求一次微分，的圖像向左平移。所以對(duì)求n次微分，那么得到的圖像就是向左平移，即得到。如前，我們用任意數(shù)替換正整數(shù)n。所以，我們得到正弦函數(shù)的任意次導(dǎo)數(shù)的表達(dá)式，同理我們也得到余弦函數(shù)的： （2）在得到表達(dá)式（2）之后，我們自然想，這個(gè)猜測(cè)與指數(shù)函數(shù)的結(jié)果是否保持一致。為了驗(yàn)證這個(gè)猜測(cè)，我們可以使用歐拉公式。利用表達(dá)式（1），我們可以計(jì)算得到，這與（2）式是吻合的。問(wèn)題問(wèn)題5：是什么？4的導(dǎo)數(shù)我們現(xiàn)在看看x次方的導(dǎo)數(shù)。我們以為例有：表達(dá)式（3）用連乘的分子和分母去替換，則得到結(jié)果如下上式就是的一般表達(dá)式。我們通過(guò)伽瑪函數(shù)，用任意數(shù)替換正整數(shù)n。當(dāng)（4）式中的p和n是不是自然數(shù)時(shí)，伽瑪函數(shù)使他們?cè)谔鎿Q后任然有意義。伽馬函數(shù)是歐拉在18世紀(jì)引進(jìn)的概念。當(dāng)時(shí)是推廣記號(hào)，當(dāng)z不是整數(shù)時(shí)。它的定義是，它具有這樣的性質(zhì)。那么我們可以將表達(dá)式（4）重新寫作這使得當(dāng)n不是整數(shù)式，（4）式還是有意義的。所以對(duì)于任意的，我們寫作利用（5）式，我們可以將分?jǐn)?shù)階導(dǎo)數(shù)延伸到很多的函數(shù)。因?yàn)閷?duì)于任意給定的函數(shù)，我們可以利用Taylor級(jí)數(shù)展開(kāi)成多項(xiàng)式的形式，假設(shè)我們可以對(duì)進(jìn)行任意次微分，那么我們得到最終那個(gè)表達(dá)式（6）呈現(xiàn)出具有作為分?jǐn)?shù)階導(dǎo)數(shù)定義候選項(xiàng)的氣質(zhì)。因?yàn)榇罅康暮瘮?shù)都可以利用Taylor公式展開(kāi)成冪級(jí)數(shù)的形式。然后，我們很快會(huì)發(fā)現(xiàn)它會(huì)導(dǎo)致矛盾的產(chǎn)生。 問(wèn)題問(wèn)題6：是否有幾何意義？5一個(gè)神秘的矛盾我們將的分?jǐn)?shù)階導(dǎo)數(shù)寫為現(xiàn)在讓我們拿它與（6）式進(jìn)行對(duì)比，看看他們是否一致。從Taylor級(jí)數(shù)來(lái)看，結(jié)合（6）式，我們得到如下表達(dá)式但是，（7）及（8）是不等價(jià)的，除非是整數(shù)。當(dāng)是整數(shù)時(shí)，（8）式的右側(cè)是的級(jí)數(shù)形式，只是用不同的表達(dá)方式。但是當(dāng)不是整數(shù)時(shí)，我們得到兩個(gè)完全不一樣的函數(shù)。我們發(fā)現(xiàn)了歷史上引起大問(wèn)題的矛盾。這看起來(lái)好像我們，指數(shù)函數(shù)的分?jǐn)?shù)階導(dǎo)數(shù)的表達(dá)式（1）與次方函數(shù)的分?jǐn)?shù)階導(dǎo)數(shù)的公式（6）是相互矛盾。正是因?yàn)橛羞@樣一個(gè)矛盾，所以分?jǐn)?shù)階微積分一般不會(huì)出現(xiàn)在初等階段的教科書里面。在傳統(tǒng)的微積分中，導(dǎo)數(shù)的次數(shù)是整數(shù)次的，求導(dǎo)的函數(shù)是初等函數(shù)。不幸的是，在分?jǐn)?shù)階微積分中，這是不正確的。通常，一個(gè)初等函數(shù)的分?jǐn)?shù)階導(dǎo)數(shù)是較高級(jí)的超越函數(shù)。關(guān)于分?jǐn)?shù)階導(dǎo)數(shù)的表格，請(qǐng)參閱文獻(xiàn)3。此時(shí)，您可能會(huì)問(wèn)我們?cè)趺蠢^續(xù)探究呢？這個(gè)謎團(tuán)將在之后的部分中被解決。敬請(qǐng)關(guān)注6多重迭代積分 我們一直在談?wù)搶?dǎo)數(shù)。積分也是反復(fù)被提及的。我們可以寫，但是等式右邊是不確定的。我們可以寫作。第二次積分可以寫成。積分區(qū)域是圖1中的三角形。如果我們交換積分的順序，那么圖1的右側(cè)圖可以表現(xiàn)出。因?yàn)椴皇且粋€(gè)關(guān)于的函數(shù)，所以可以將里面的積分移到外面，即或者。使用相同的過(guò)程步驟，我們可以寫出在一般情況下，現(xiàn)在，我們用先前做的方法，用任意數(shù)替換，用伽瑪函數(shù)替換階乘，然后得到這個(gè)一般性的表達(dá)式（使用積分）的分?jǐn)?shù)階導(dǎo)數(shù)表達(dá)式，有成為定義的潛力。但是存在一個(gè)問(wèn)題。如果該積分是反常積分。因?yàn)楫?dāng)對(duì)任意，積分是發(fā)散的。當(dāng)反常積分收斂。所以當(dāng)是負(fù)數(shù)時(shí)，原表達(dá)式是正確的。因此當(dāng)是負(fù)數(shù)時(shí)（9）式收斂，即它是一個(gè)分?jǐn)?shù)階次積分。在我們結(jié)束這一部分之前，需要提下，趨于零的下極限是任意的?？梢院?jiǎn)單的認(rèn)為存在下極限。但是會(huì)造成最后結(jié)果表達(dá)式的不同。正因?yàn)槿绱耍芏噙@個(gè)領(lǐng)域的研究人員使用符號(hào)。這個(gè)符號(hào)說(shuō)明了極限過(guò)程是從到的。這樣我們從（9）式得到問(wèn)題問(wèn)題7：如下分?jǐn)?shù)階微分的下極限是什么？7解秘現(xiàn)在，你可以開(kāi)始去發(fā)現(xiàn)前面哪些地方出錯(cuò)了。我們對(duì)于分?jǐn)?shù)階積分包含極限，并不感到驚訝。因?yàn)榉e分是涉及到極限的。然而普通的導(dǎo)數(shù)不涉及積分的極限，沒(méi)有人希望分?jǐn)?shù)階導(dǎo)數(shù)包含這樣的極限。我們認(rèn)為，導(dǎo)數(shù)是函數(shù)的局部性質(zhì)。分?jǐn)?shù)階導(dǎo)數(shù)的符號(hào)既包含導(dǎo)數(shù)（是正數(shù)）又有積分（是負(fù)數(shù)）。積分是處于極限之中的。事實(shí)證明，分?jǐn)?shù)階導(dǎo)數(shù)也是處于極限之中的。出現(xiàn)該對(duì)矛盾的原因是，我們使用了兩種不同的極限。現(xiàn)在，我們可以解決這個(gè)謎團(tuán)了。秘密是什么？讓我們停下來(lái)想一想。表達(dá)式（1）中指數(shù)函數(shù)起作用的極限是什么？記得我們要期望寫成 b取什么值時(shí)，將得到這個(gè)答案？由于在（11）式中積分就是為了得到我們想要的形式，只有當(dāng)時(shí)。即如果a是正數(shù)，那么。這種類型的擁有下極限為的積分，有時(shí)也稱為Weyl分?jǐn)?shù)階導(dǎo)數(shù)。從（10）的符號(hào)，我們可以將（1）寫做極限在公式（5）的導(dǎo)數(shù)中是起什么作用？我們有 同樣，我們希望。當(dāng)時(shí)，結(jié)論是成立的。所以我們覺(jué)得將（5）的符號(hào)寫成更準(zhǔn)確。因此，表達(dá)式（5）中隱含了的下極限為0。然而，表達(dá)式（1）中的下極限為。這個(gè)差異就是（7）和（8）為什么不等價(jià)的原因。在（7）中我們計(jì)算，在（8）中我們計(jì)算。如果讀者希望繼續(xù)這一研究，我們推薦Miller的一篇很好的論文8，和由Oldham和Spanie合著的優(yōu)秀圖書11，以及Miller和Ross合著的優(yōu)秀圖書9。這兩本書都包含了從很多文獻(xiàn)角度分?jǐn)?shù)階微積分簡(jiǎn)短精要的歷史。由Miller和Ross合著的圖書9很好地討論了分?jǐn)?shù)階微分方程。Wheeler的注記14是另一個(gè)這方面一流的介紹性文章，具有很高的推廣普及價(jià)值。Wheeler給出了幾個(gè)簡(jiǎn)單的應(yīng)用例子，而且閱讀起來(lái)非常有趣。其他的歷史性文獻(xiàn)請(qǐng)參考1，2，4，5，6，10，13。8問(wèn)題的解答以下是文章中8個(gè)問(wèn)題的簡(jiǎn)短回答。問(wèn)題1：成立的，這個(gè)性質(zhì)是保持的。問(wèn)題2：成立的，這個(gè)由關(guān)系式（2.2）很容易證明。問(wèn)題3：遺漏了一些東西。遺漏的是積分常數(shù)。應(yīng)該是這樣的問(wèn)題4：將展開(kāi)成傅立葉級(jí)數(shù)形式，。假設(shè)我們可以連續(xù)次地取分?jǐn)?shù)階微分，我們得到。問(wèn)題5： 。問(wèn)題6：我們知道表示的曲線斜率，表示曲線的凹凸性。但三階和更高階導(dǎo)數(shù)給出的幾何意義很少或根本沒(méi)有。那么對(duì)于這些特殊的導(dǎo)數(shù)，分?jǐn)?shù)階導(dǎo)數(shù)沒(méi)有簡(jiǎn)單的幾何意義對(duì)我們來(lái)說(shuō)也并不感到驚訝了。 問(wèn)題7：分?jǐn)?shù)階微分的下極限是。 【參考文獻(xiàn)】1 A. K. Grunwald, Uber begrenzte Derivationen und deren Anwendung, Z. Angew.Math. Phys., 12 (1867), 441480.2O. Heaviside, Electromagnetic Theory, vol. 2, Dover, 1950, Chap. 7, 8.3J. L. Lavoie, T. J. Osler and R. Trembley, Fractional derivatives and special functions, SIAM Rev., 18 (1976), 240268.4A. V. Letnikov, Theory of differentiation of fractional order, Mat. Sb., 3 (1868),168.5J. Liouville, Memoire sur quelques questions de gometrie et de mcanique, et su run noveau gentre pour resoudre ces questions, J. cole Polytech., 13(1832), 169.6J. Liouville, Memoire: sur le calcul des differentielles indices quelconques, J.cole Polytech., 13(1832), 71162.7A. C. McBride and G. F. Roach, Fractional Calculus, Pitman, 1985.8K. S. Miller Derivatives of noninteger order, Math. Mag., 68 (1995), 183192.9K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, 1993.10P. A. Nekrassov, General differentiation, Mat. Sb., 14 (1888), 45168.11K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, 1974.12T. J. Osler, Fractional derivatives and the Leibniz rule, Amer. Math. Monthly, 78 (1971), 645649.13E. L. Post, Generalized differentiation, Trans. Amer. Math. Soc., 32 (1930) 723781.14B. Ross, editor, Proceedings of the International Conference on Fractional Calculusand its applications, Springer-Verlag, 1975.15N. Wheeler, Construction and Physical Application of the fractional Calculus, notes for a Reed College Physics Seminar, 1997.原文： A Childs Garden of Fractional DerivativesMarcia Kleinz and Thomas J. OslerThe College Mathematics Journal, March 2000, Volume 31, Number 2, pp. 8288Marcia 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 () 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.IntroductionWe are all familiar with the idea of derivatives. The usual notationor ， or is easily understood. We are also familiar with properties likeBut what would be the meaning of notation like or ？Most readers will not 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 LHospital, 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 definitionlemmatheorem approach, we explore the idea of a fractional derivative 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 functionsWe 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 complex number like1+i ?We will be bold and write (1)for any value of , integer, rational, irrational, or complex. 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.QuestionsQ1 In this case does ?Q2 In this case does ?Q3 Is , and is ,(as listed above) really true, oris there something missing?Q4 What general class of functions could be differentiated fractionally be means ofthe 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 Eulers expression, Using (1) we can calculatewhich agrees with (2).QuestionQ5 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 (p-n)! results inThis 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 (p-n)! 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 non-integer values of z. Its definition is ,and it has the property that .We can rewrite (4) aswhich makes sense if n is not an integer, so we putfor any . With (5) we can extend the idea of a fractional derivative to a large number offunctions. Given any function that can be expanded in a Taylor series in powers of x,assuming we can differentiate term by term we getThe 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 Taylors series in powers of x. However, we will soon see that it leads to contradictions.QuestionQ6 Is there a meaning for in geometric terms?A mysterious contradictionWe wrote the fractional derivative of asLet us now compare this with (6) to see if they agree. From the Taylor Series, (6) givesBut (7) and (8) do not match unless is a whole number! When is a whole number, the right side of (8) will be the series of with different indexing. But when is not a whole number, we have two 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 integralsWe have been talking about repeated derivatives. Integrals can also be repeated. We could write ,but the right-hand side is indefinite. We will instead write .The second integral will then be .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 thatSince is not a function of , it can be moved outside the inner integral so,orUsing the same procedure we can show thatand, in general,Now, as we have previously done, let us replace the n with arbitraryand the factorial with the gamma function to getThis 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 as The integral diverges for every .When the improper 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)QuestionQ7 What lower limit of fractional differentiation b will give us the result?The mystery solvedNow 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 local 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? Lets stop and think. What are the limits that will work for theexponential from (1)? Remember we want to writeWhat value of b will give this answer? Since the integral in (11) is reallywe will get the form we want when .It will be zero when So, if a is positive, 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) asNow, what limits will work for the derivative of in (5)? We haveAgain we want 。This will be the case when. We conclude that (5) should be written in the more revealing notationSo, the expression (5) for has a built-in lower limit of 0. However, expression (1) for hasas a lower limit. This discrepancy is why (7) and (8) do not match. In(7) we calculated and in (8) we calculated.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. Wheelers 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 questionsThe 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 haveQ4 Let be expandable in an exponential Fourier series, . Assuming we can differentiate fractionally term by term we get Q5 Q6 We know thatis geometrically interpreted as the slope of the curveand gives us the concavity of the c