PHGN311 Homework #2
Due Friday, Sep. 6, 2013 at the beginning of class
Show your work on all problems. You can use something like Mathematica to check
your results, but don’t turn it in as the solution unless asked to.
We are nearly done reviewing series, so begin reading Chapter 2 on complex numbers
which is likely a review of things you have seen.
1. Boas 1.9.22
2. Boas 1.10.16
3. Boas 1.10.23
4. Boas 1.13.4 You can use Mathematica to check the answer, but be sure to show your
actual work
5. Boas 1.13.14
6. Consider the function f(x)= 1/(3x-1)
a. Obtain a power series expansion about x=2 for f (something of the form
∑ an (x −1 2)n ).
b. Show that the x range for which the power series converges is 1/3 < x < 11/3
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c. Using your favorite computational tool, plot f for 1/3 < x < 11/3 and on the
same graph plot partial sums of the power series representation of f including
terms up to n=2, n=11, n=20, n=101.
d. Notice that convergence is best near x=2, and that the error alternates in sign
near x= 11/3 but monotonically approaches the limiting value near x=1/3.
Why is this?
e. Now, obtain a Laurent series expansion for f about x=2 (something of the
an
form ∑
) which converges for x>11/3 and x< 1/3.
(x − 2)n
f. Plot f for -2 < x < 1/3 and on the same graph plot partials sums of the Laurent
series that include the first 3 terms, first 12 terms, and first 21 terms.
7. Boas 1.15.28
8. Boas 1.16.28