MATH 348 - Advanced Engineering Mathematics February 22, 2008
Chapter 11.1-11.4 Q+A, Spring 2008
Response to feedback from Homework 3 - Question 5
1. Personal Recap - At this point we will pause our work with Fourier series and thus it is a good time to summarize
our recent developments. Please do the following for each of the sections 11.1, 11.2, 11.3, 11.4:
(a) In roughly one paragraph summarize the section. Pretend you are trying to explain what was missed to
a friend of yours who missed class. Your goal should be to recount key concepts and connect them to the
important mathematics encountered.1
(b) List one to three questions remaining with the material.
Q: When does a0= 0?
A: a0= 0 can happen when the periodic function is odd or when Zπ
−π
f(x)dx = 0 for a 2π-periodic function.
Q: When do you use a half-wave rectifier?
A: http://en.wikipedia.org/wiki/Rectifier - The first paragraph gives great examples.
Q: When would you choose an even extension?
A: If you have the option of periodically extending a function then you might as well extend the function in
a symmetric way. If this is the case then it is unclear whether to choose an odd or even extension. If you
want the extended function to be smooth then you should choose an even extension. However, this requires
two integral evaluations. If you are not concerned about smoothness then odd extensions will work as well.
Q: When would you use a complex Fourier series?
A: There is no choice mechanism for which you would use, real or complex, otherwise only one would be
discussed in books and class. The choice mechanism basically runs like this. If you want to simplify your
coefficient calculations and are not concerned with having imaginary numbers then you would use a complex
FS. If you do not want complex numbers then you would use a real FS.
Q: How do you convert a complex FS to a real FS.
A: The process has three parts. First, rationalize your cncoefficients. After this write the doubly infinite sum
as two infinite sums one over the positive integers and one over the negative integers. Finally, join the two
sums. If everything has been done correctly then all imaginary terms drop out.
Q: Can you go from a real FS to complex FS?
A: Yes, we did the derivation in class. The derivation is also done in 11.4 of the text.
Q: Why use a FS to estimate a function when we know the function?
A: This is a great question. Say you are given a function f(x) = xwhich is 2π-periodic on the interval
−π < x < π. Well, what is f(100)? This is difficult to answer in the current form of fhowever, if you
knew the FS then you could determine this value, within some precision, pretty easily from the FS of f.
1Please be concise. The key is to strip the material down to what you feel is essential. If you write too much then you are likely to
forget much of the summary.
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