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原文：
A Child’s Garden of Fractional Derivatives
Marcia Kleinz and Thomas J. Osler
Introduction
We are all familiar with the idea of derivatives. The usual notation
or ， or
is easily understood. We are also familiar with properties like
But what would be the meaning of notation like or ？Most readers will not have encountered a derivative of “order ” before, becau almost none of the familiar textbooks mention it. Yet the notion was discusd 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 ide
a. 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 t of very readable minar notes has been prepared by Wheeler [15], but the have not beenpublished.
It is the purpo 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 looking at examples of familiar n th order derivatives like and then replacing the natural number n by other numbers like In this way, like detectives, we will try to e 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 e the excellent expository paper by Miller [8].)
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As the exploration continues, we will at times ask the reader to ponder certain questions. The answers to the questions are found in the last ction of this paper. So just what is a fractional derivative? Let us e. . . .
Fractional derivatives of exponential functions
We will begin by examining the derivatives of the exponential function becau the patterns they develop lend themlves 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 n th iterated integral. reprents 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 ud this approach to fractional differentiation in his papers [5] and [6]. Questions
Q1 In this ca does ?
Q2 In this ca does ?
propos
Q3 Is , and is ,(as listed above) really true, or
凝聚的意思
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 prents no obvious pattern from which to find . However, graphing the functions disclos 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 the guess are consistent with the results of the previous ction for the exponential. For this purpo we can u Euler’s expression,
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Using (1) we can calculate
which agrees with (2).
Question
Q5 What is ?
木瓜什么意思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 in
This is a general expression of .To replace the positive integer n by the arbitrary number we may u 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) as
which makes n 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 ofmisty>i
elfunctions. Given any function that can be expanded in a Taylor ries in powers of x,
assuming we can differentiate term by term we get
The final expression prents itlf 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 ries in powers of x. However, we will soon e that it leads to contradictions.
Question
Q6 Is there a meaning for in geometric terms?
A mysterious contradiction
We wrote the fractional derivative of as
Let us now compare this with (6) to e if they agree. From the Taylor Series, (6) gives
But (7) and (8) do not match unless is a whole number! When is a whole number, the right side of (8) will be the ries of with different indexing. But when is not a whole number, we have two entirely different functions. We have discovered acontradiction that historically has caud 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 e [3].
At this point you may be asking what is going on? The mystery will be solved in later ctions. Stay tuned . . . .
Iterated integrals
We 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 cond 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 that
Since is not a function of , it can be moved outside the inner integral so,
petition
or
Using the same procedure we can show that
and, in general,
Now, as we have previously done, let us replace the –n with arbitraryand the factorial with the gamma function to get
This is a general expression (using an integral) for fractional derivatives that has the potential of being ud as a definition. But there is a problem. If the
integer is improper. This occurs becau 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 ction 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. Becau of this, many people who work in this field u the notation indicating limits of integration going from b to x. Thus we have from (9)
Question
Q7 What lower limit of fractional differentiation b will give us the result
?
The mystery solved
Now you may begin to e what went wrong before. We are not surprid that fractional integrals involve limits, becau 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 ud. Now we can resolve the mystery.
What is the cret? Let’s stop and think. What are the limits that will work for the
exponential from (1)? Remember we want to write
What value of b will give this answer? Since the integral in (11) is really
we 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) as
Now, what limits will work for the derivative of in (5)? We have
摘要英文翻译
Again we want 。This will be the ca when. We conclude that (5) should be written in the more revealing notation
So, 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
