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PHYCS 3730/6720 Lab Exercise

References:

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Exercise 1.

Bessel functions appear commonly in problems with cylindrical
geometry. For example the radial profile of the normal vibrational
modes of a drumhead is described by Bessel functions. Bessel
functions come with an integer index **J**_{n}(x)
where **n** = 0, 1, .... In Maple notation they
are **BesselJ(n,x)**.
Use Maple to plot the Bessel function **J**_{1}(x) for a
range sufficient to show the first three positive roots (not counting the one
at **x = 0**). In the answer file **Mylab06.txt**, give an estimate of the zeros, based on
reading the plot.

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Exercise 2.

Use Maple to develop the Taylor/MacLaurin series representation of the
first order Bessel function **J**_{1}(x) up to and
including the term in
**x^9**. Note, the **O** means "order of". So
**O(x**^{11}) means the rest of the series vanishes with
**x** as **x**^{11}.
For help with the Taylor/MacLaurin series in Maple, use
**?taylor**. The **J**_{m}(x) Bessel functions are called
**BesselJ(m,x)**. Use **?BesselJ** to learn more and see how to
get related Bessel functions.
When you are satisfied with your result, copy your answer from the
Maple window to the file **Mylab06.txt**. Please provide both input and
output lines for the results you want to hand in.

The simplest way to do this is to use your mouse to copy from the
Maple window and paste it into your emacs window.

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Exercise 3.

With Maple use the Newton-Raphson method to find the zero of the
**J**_{1}(x) Bessel function near x = 4 to a tolerance of
10^{-4}.
Hint: For this exercise you need to evaluate the derivative of the
Bessel function. Maple insists on being much more precise about this
procedure than most scientists are used to. First you have to find
the function that is the derivative of the Bessel function. Then you
have to evaluate the derivative function at the desired point. There
are two ways to do this in Maple, namely, with **diff** or
**D** for the derivative.

f := x -> BesselJ(1,x); g := x -> evalf( D(f)(x) );

or
f := x -> BesselJ(1,x); g := x -> evalf( subs(y=x, diff(BesselJ(1,y),y) ) );

You don't need to write a Maple procedure for this. Use the same
technique you used for the fixed point method, i.e. write an
assignment statement that gives the new Newton Raphson estimate
of **x** based on the old estimate. Then repeat by hand until you
are satisfied that it has converged.
In the answer file, show the Maple commands you used and the result
you found.

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Exercise 4.

Use the Maple **fsolve** command to find the same root.
Copy the command and your answer to the answer file.
Note that with **fsolve**, you can specify a range of **x**
values in which to search for the root. Check it out with
**?fsolve**.