A note about the sinc function as a solution to a differential equation

The (infamous , or famous , take your pick) sinc function is known to be one of two solutions of the differential equation:

differential equation with sinc function as a solution

This is a linear second order ordinary differential equation with dependent variable x and independent variable y.
I have tried to explore and find the solutions to this differential equation using mostly computer math software and programs.

The Texas Instruments 92 Plus scientific calculator and the Maple computer algebra system agree and give the same solution :

sinc solution to differential equationMathematica gives the following solution:

solution with expAfter converting the exponentials to trigonometric functions the expression above and the solution of the differential equation given by Mathematica becomes:

 Mathematica sinc solution

This solution is less simple than (1). Note that if we make the assumption in Mathematica that λ > 0 ,we get the solution:

solution with lambda positiveIf we try to find the graphical representation of solution (1) for different values of the arbitrary constants c1 and c2 and with λ (or n ) between -3 and 3 ,we get the following graphs:

sinc 2d graphs solution

Then I tried to take the absolute value of λ in the numerator of (2) for different values of the arbitrary constants , and compared their graphs  with the graphs of solution (1) for the same values of the constants. λ in (2) and n in (1) are between 1 and 4.graphs of sinc solution of differential equation

The graphical results are similar but not exactly the same.

Using the Manipulate built-in function in Mathematica , here is an animation of the graphs for solution (1) obtained by varying λ , c1 and c2:

Finally , I will give the solution of a generalized form of the differential equation above

general form of differential equation

obtained with Mathematica :

solution of generalized differential equationIn the solution above , Jμ(x) is the Bessel function of the first kind , and Yμ(x) is the Bessel function of the second kind.

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Some more 3D graphics related to the sinc function

I’ll finish exploring the sinc function by showing a few 3D graphs of sinc related curves.
Here is a first set of 3D curves of sinc related functions:

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As an example , here is the Mathematica code for the 3D curve √(x² + y²)sinc(√(x² + y²)) above:

sinc code sqrt

And here is the second set of 3D surfaces related to sinc:

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The Mathematica code for the curve given by sinc(x² -y²) is:

sinc 3d surface squared

I’m somewhat fed up with sinc , so soon I will move on to other subjects.

Update: I will add a last group of six 3D curves related to sinc . They include curves where sinc as a function of x is multiplied by sinc as a function of y , such as sinc(x)×sinc(y) , sinc(ln(x))×sinc(ln(y)) with ‘ln’ the natural logarithm to the base e  rendered as ‘log’ by Mathematica in the image , sinc(sin(x))×sinc(cos(y)) , and sinc(x²)×sinc(y²)  .

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2D and 3D graphics related to the sinc function

After having explored calculus results related to the sinc function , here are a few 2D and 3D graphic results.

I’ll start with the integral :

integral of sinc

The graph of the function above is (made with Mathematica):

sinc functionWe’ll consider more than one way to graph this function in 3D.
First , by taking a specific function of x and y and using the following Mathematica code :

 sinc-sq
We get the 3D result :

sinc 3 d function
We can use the RevolutionPlot3D buit-in function in Mathematica to get the graph of the solid obtained from the function by revolving around the z-axis:

solid around z-axis code

sinc function z-axis
The graph of the solid obtained by revolving about the x-axis is:

sinc function x-axis
The Mathematica code for the graph above is:

code for solid x-axis
And the graph of the solid obtained from the function by revolving about the y-axis is:

3d graph about y-axis
The Mathematica code for the graph above is:

code for graph about y-axis
Next is a comparison  between the 2D graph and the 3D graph of the function sinc(tan(w)):

sinc tan
Below is a  set of six 2D graphs of sinc as a function of hyperbolic trigonometric functions:

2d sinc hyperbolic trigMathematica code for the graphs above( rd will be used in the 3D graphs later):

sinc 2d hyperbolic trig

Here are the six 3D graphs related to the above six 2D sinc graphs:

group of 3d sinc functions  trigonometric

And the Mathematica code for the 3D sinc graphs above:

sinc-3d-test-gen

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