# A few calculus results related to the sinc function

I’ve tried to sink my teeth into the sinc function and obtained the following calculus solutions , mostly by tinkering with Mathematica.

The sinc function is generally defined by:

$\text {sinc} (x) = \frac {\sin (x)} {x}$

with sinc(0) = 1.
The sinc function is sometimes called the filtering or interpolation function and is often used in digital signal processing and in engineering . Sometimes a distinction is made between the unnormalized sinc function and the normalized sinc function sin(πx)/(πx) , but I’m going to consider mostly the unnormalized function.

Graph of the sinc function (done with Mathematica):

Here is a table of the jth derivative of sinc(x) for j between 1 and 6:

And for j between 7 and 10:

Another table of derivatives for sinc of x with x to the nth power:

Also :

Si(x) is the sine integral function .

Here is an indefinite integral of sinc(f(x)) with

$f(x)=x^n$

For the definite integral we get:

Another indefinite integral :

Ei(x) is the exponential integral function .

And below is a table of values for a definite integral of sinc(x) to the jth power: