Math 129 Section 005H
For Your Information: an application of the gamma function

On the last homework we learned about the gamma function

$$\mathrm{\Gamma }(x)={\int }_0^{\mathrm{\infty }}{t}^{x-1}{e}^{-t}𝑑t.$$

It has many uses, one of which is to provide a continuous analog of the factorial. This is because $\mathrm{\Gamma }$ is a continuous function for $x>0$ and satisfies $\mathrm{\Gamma }(n)=(n-1)!$ for all positive integers. In this note, I’ll discuss one example.

The example involves the (somewhat miraculous) formula

$$\frac{d^n}{d{x}^n}{\int }_0^x\frac{{(x-y)}^{n-1}}{(n-1)!}f(y)𝑑y=f(x).$$

That is, the function

$$F^{(n)}(x)={\int }_0^x\frac{{(x-y)}^{n-1}}{(n-1)!}f(y)𝑑y$$

is an $n$th antiderivative of $f$! (It is the unique antiderivative whose value and first $n-1$ derivatives at $x=0$ are all zero.)

For example, if we plug in $n=1$, we get

$$\frac{d}{dx}{\int }_0^{x}f(y)𝑑y=f(x),$$

which is the Fundamental Theorem of Calculus. If we plug in $n=2$, we get

$$\frac{d^2}{d{x}^2}{\int }_0^x(x-y)f(y)𝑑y=f(x),$$

and so on.

Using the Gamma function, we can generalize the above to non-integer values of $n$, i.e., if $r$ is any positive real number, define

$$F^{(r)}(x)={\int }_0^x\frac{{(x-y)}^{r-1}}{\mathrm{\Gamma }(r)}f(y)𝑑y$$

For non-integer values of $r$, this defines a fractional integral of $f$ of order $r$. Among other things, it satisfies

$$\frac{d^n}{d{x}^n}{F}^{(r)}(x)=F^{(r-n)}(x)$$

for any positive real number $r$ and any nonnegative integer $n\le r$.