【正文】 and [6]. Questions Q1 In this case does 1 2 1 21 2 1 2()a x a x a x a xD c e c e c D e c D e? ? ? ?? Q2 In this case does ax axD D e D e? ? ? ??? ? Q3 Is 1 ()ax axD e e dx? ?? , and is 2 ()a x a xD e e dx dx? ? ?? ,(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: 0 1 2sin sin , sin c o s , sin sin ,D x x D x x D x x? ? ? ? 10 This presents no obvious pattern from which to find 1/2sinDx. However, graphing the functions discloses a pattern. Each time we differentiate, the graph of sin x is shifted /2? to the left. Thus differentiating sin x n times results in the graph of sin x being shifted /2n? to the left and so sin sin ( )2n nD x x ???. 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: s in s in ( ) , c o s c o s ( ) .22D x x D x x?? ?? ??? ? ? ? (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, cos si nixe x i x?? Using (1) we can calculate ( / 2 ) c o s ( ) s in ( )22i x i x i i xD e i e e e x i x? ? ? ? ?? ??? ? ? ? ? ? which agrees with (2). Question Q5 What is sin( )D ax? ? Derivatives of px We now look at derivatives of powers of x. Starting with px we have: 0 1 2, , ( 1 ) , ,( 1 ) ( 2 ) ( 1 ) . ( 3 )p p p p p pn p p nD x x D x p x D x p p xD x p p p p n x?? ? ? ?? ? ? ? ? Multiplying the numerator and denominator of (3) by (pn)! results in ( 1 ) ( 2 ) ( 1 ) ( ) ( 1 ) 1 ! ( 4 )( ) ( 1 ) 1 ( ) !p p n p np p p p n p n p n px x xp n p n p n??? ? ? ? ? ? ???? ? ? ? This is a general expression of npDx .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 10( ) dtzz e t t? ????? ,and it has 11 the property that ( +1) !zz??. We can rewrite (4) as ( 1) ,( 1)n p p npD x xpn ???? ? ? ? which makes sense if n is not an integer, so we put ( 1 ) ( 5 )( 1 )pp pD x xp?? ? ???? ? ? ? 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, 0( ) ,nnnf x a x???? assuming we can differentiate term by term we get 00( 1 )( ) . ( 6 )( 1 )nnnnnnnD f x a D x a xn? ? ???? ??????? ? ? ??? 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 ()D f x? in geometric terms? A mysterious contradiction We wrote the fractional derivative of as ( 7 )xxD e e? ? Let us now pare this with (6) to see if they agree. From the Taylor Series, 01 ,!xnnexn????(6) gives 0 . ( 8 )( 1 )nxnxDe n? ???? ? ? ?? 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 series of xe 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 12 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 ( ) ( )D f x f x d x? ? ? ,but the righthand side is indefinite. We will instead write 1 0( ) ( )xD f x f t d t? ? ? .The second integral will then be 22 1 1 200( ) ( )xtD f x f t d t d t? ? ?? . The region of integration is the triangle in Figure 1. If we interchange the order of integration, the righthand diagram in Figure 1 shows that 12 1 2 10( ) ( )xxtD f x f t d t d t? ? ?? Since 1()ft is not a function of 2t , it can