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Published 2000 Revised 2019
There has been a lot of correspondence recently on the Ask Nrich web-board about fractional derivatives. We know how to differentiate a function once, twice and so on, but can we differentiate the function 3/2 times? Similarly, we know how to integrate a function once, twice, and so on, but can we integrate it 1/2 times? This is the first of three articles which will introduce you to the main ideas in this new and rather strange world of fractional integrals and derivatives. Questions about the existence of such things were asked not long after calculus was created; for example, in 1695 Leibnitz wrote "Thus it follows that d^{1/2}x will be equal to \ldots from which one day useful consequences will be drawn ." Also, Euler (1738) wrote "When n is an integer, the ratio d^np, p a function of x, to dx^n can always be expressed algebraically. Now it is asked: what kind of ratio can be made if n is a fraction?"
If we differentiate x^n n times, where n is a positive integer, we get n! ; thus {d^n\over dx^n}x^n = n!\ .
Let F(n) be the factorial function; then for every positive integer n we have F(n) = 1.2.3\cdots n. Of course, we usually write n! instead of F(n), and we can define F(n) (and therefore n!) by the conditions F(1)=1, \quad F(n) = nF(n-1), \quad n=2,3,\ldots. \quad (1.1)
In order to define y! for every positive y we need to discuss an extremely important function in mathematics known as the Gamma function \Gamma(x). This function is defined for every positive x, and it satisfies the intruiging formulae \Gamma(1)=1, \quad \Gamma(x+1) = x\Gamma(x), \quad x> 0. \quad (1.2)
Next, if we write G(x) = \Gamma(x+1) we obtain G(1)=1, \quad G(x) = xG(x-1),
For the moment, we continue with our discussion of the consequences of (1.2) and (1.3) even though we have still not defined the Gamma function. First we note that (1.3) gives the curious formula 0! = \Gamma(1)=1.
Speaking of binomial coeffients, we recall that if k and n are positive integers with n> k, then the corresponding binomial coefficient is defined to be {n\choose k} = {n!\over k!\,(n-k)!}.
Binomial coefficients with non-integral entries are used in the Binomial expansion with non-integral powers. For example, we have the Binomial Theorem (1+x)^n = \sum_{k=0}^n {n\choose k}x^k
Quite generally, if n is a positive integer and 0\leq\theta \leq 1 , from (1.2) and (1.3) we have (n+\theta)! = \Gamma(n+\theta +1) =(n+\theta)(n-1+\theta)\cdots (1+\theta)\theta\Gamma(\theta).
All of this discussion depends on having the Gamma function available so how, then, do we define the Gamma function? The definition is this : \Gamma (x) = \int_0^\infty t^{x-1}e^{-t}\,dt.
The formula \Gamma(x+1) =x\Gamma(x) enables us to calculate \Gamma(x) in terms of the value of \Gamma at the fractional part of x. The following illustrative example will show what we mean here : (7/2)! = \Gamma (9/2) = {7\over 2}\Gamma(7/2) = {7\over 2}{5\over 2}\Gamma(5/2) = \cdots = {7\over 2}{5\over 2}{3\over 2}{1\over 2}\Gamma (1/2) ={105\over 16}\Gamma(1/2),
The values of the Gamma function are given in tables (just as the values of \sin x are), and using these tables we can calculate, for example, (9/4)!: its value is \Gamma(13/4) = 2.5493\cdots.
The following table of values of \Gamma(0.1),\ldots ,\Gamma(0.9) will enable you to find some values of y!
k | \Gamma(k/10) |
---|---|
1 | 9.5135 |
2 | 4.5908 |
3 | 2.9916 |
4 | 2.2182 |
5 | 1.7725 |
6 | 1.4892 |
7 | 1.2981 |
8 | 1.1642 |
9 | 1.0686 |
Using this you should be able to see, for example, that (3/2)! = 1.3293, \quad (3.7)! = 15.431,\quad (4.2)! = 32.578, \quad (5.2)! = 169.41, \quad (6.7)! = 2769.8.
Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?
Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x
Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.