Fractional calculus I
Introduction
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
The Factorial function
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
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
Next, if we write $G(x) = \Gamma(x+1)$ we obtain
Binomial coefficients
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
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
Binomial coefficients with non-integral entries are used in the Binomial expansion with non-integral powers. For example, we have the Binomial Theorem
Quite generally, if $n$ is a positive integer and 0\leq\theta \leq 1 , from (1.2) and (1.3) we have
The Gamma function
All of this discussion depends on having the Gamma function available so how, then, do we define the Gamma function? The definition is this :
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 :
Evaluating the Gamma function
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