Fourier-Bessel series

Orthogonality of Bessel functions

This is what the Bessel function looks like.

We see there are zeros of the function at x ∼ 2.4, 5.5, 8.7 ...

Orthogonality

The orthogonality integral is (where a and b are zeros of the Bessel function)

The FullSimplify command will convert negative order Bessel functions.

The functions BesselJ[0,b]and BesselJ[0,a] are zero since a and b are zeros of the functions.  So the integral will vanish.  But if a - b then denominator will go to zero as well so must look further.  Evaluate this integral separately:

First term is zero.  Second term is not; this gives the normalisation function.

This establishes the orthogonality and normalisation of the where α are the zeros of the Bessel function.

Tabulating the zeros

To proceed we neet to have the zeros of available.  These may be found using the NumericalMath package.  First we must load this in:

Next we shall construct a list of the first 50 zeros.  We will call this list "zeros".

Let's just test to see that the usual list operation will extract the element:

That's OK.  So now we define the function α[n] which gives the zero.

Let's just test this.

Seems OK.  Actually, we don't really need this function; we could regard zeros[[n]] as the function.

First term is zero.  Second term is not; this gives the normalisation function.

Let's just test that these really are the zeros of the Bessel function

This is very small. The Chop function removes small residuals like this, returning zero, as expected.

Some examples

Cos function

First we define the function.

Now let's plot it:

This is the integral for the Bessel coefficients:

Let's test it for the coefficient.

We see that Mathematica can't do the integral.  So we must ask for a numerical evaluation.

Similarly we can obtain the second Bessel coefficient.

The sensible thing to do is to make a table of the numerical coefficients.  If we then work with the elements of this table then there are no more integrals to be done.

Now we will define the sum of the first m Bessel terms, so we can make comparisons for different m.

Here is the fundamental:

Here is the sum of the first two terms

And here are various partial sums for comparison

The next picture shows the error in the 4th partial sum

Finally we give a graphical indication of the Fourier-Bessel coefficients. We need the Graphics package for bar graphs.

Parabola

First we define the function.

Now let's plot it:

This is the integral for the Bessel coefficients

Let's test it:

Now we put the coefficients together in a table.

Now we will define the sum of the first m Bessel terms, so we can make comparisons for different m.

This is the fundamental

Here is the sum of the first two terms

And here are various partial sums for comparison

The next picture shows the error in the 4th partial sum

Finally we give a graphical indication of the Fourier-Bessel coefficients.

Box

First we plot the function

This is the integral for the Bessel coefficients

Let's test this:

Now we put the coefficients together in a table.

And this is what the coefficients look like

Now we will define the sum of the first m Bessel terms, so we can make comparisons for different m.

This is the fundamental

Here is the sum of the first two terms

And here are various partial sums for comparison

This is the tenth partial sum

Triangle

Let's define the function.

Now plot it

This is the integral for the Bessel coefficients

Now do a couple of tests:

Now we put the coefficients together in a table.

These can be plotted as a histogram

Now we will define the sum of the first m Bessel terms, so we can make comparisons for different m.

This is the fundamental

Here are the first two terms

Here is the third partial sum

Here we plot a number of partial sums for comparison.

This is the tenth partial sum; it looks pretty good

Finaly we examine the error in the 10th partial sum.

Sine cycle

Let's define the function.

Now plot it

This is the integral for the Bessel coefficients

Now let's test this:

We see that Mathematica can't do the integral.  So we must ask for a numerical evaluation.  We shall put these numerical values into a table.

Let's plot these:

Now we will define the sum of the first m Bessel terms, so we can make comparisons for different m.

This is the fundamental:

Here are the first two terms

Here are the first 4 partial sums.

This is the fifth partial sum.

And here is the error in the fifth partial sum

2 Sine cycles