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):

sinc function graph

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

sinc-deriv-1-6

And for j between 7 and 10:

sinc-deriv-7-10

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

calculus-sinc-1

 

Also :

sinc integral set 4

Si(x) is the sine integral function .

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

 f(x)=x^n

sinc-properties-int-1

 

For the definite integral we get:

sinc definite integral

Another indefinite integral :

sinc integral 2

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:

sinc integral 3

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