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:
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:
Si(x) is the sine integral function .
Here is an indefinite integral of sinc(f(x)) with
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: