Find the Fourier coefficients of the following equation and write the function as infinite series.
V = 20 + 10 sin t
so far im hovering around these values;
-5i n = 1
5i n = -1
20 n = 0
but I dont really understand what it is asking me to do when it says ' write the function as infinite series'. Am I just totally off track?
Whether you are "completely off base" depends on exactly what you think a "Fourier series" is! A Fourier series is an infinite series of the form
However, it is quite possible for most of those coefficients, and , to be 0! In this case, you don't have to do any "calculation" at all- just compare
and 20+ 10 sin(t).
The definition I have for the Fourier Series, (with as the fundamental period), is:
where is (using as the original equation from the OP):
I can use Euler's identity to get your definition, but you are starting your series at , the definition I have starts at . Why the indexing difference?
Using your definition, shouldn't for all ? If your definition comes from my definition and Euler's identity, the coefficients would be the same.
Last, I tried (for practice) the problem from the OP, using as the fundamental period). I tried comparing to your series definition. I get:
Is that right? If so, what is the final answer (the infinite Fourier Series)? What about the in original equation?
I also tried the definition I have for . But those integrations return very complicated answers (I used Maple to calculate them), and nothing like what I have in the last paragraph. Why doesn't that integration turn up the same results?
Thanks, in advance, for anyone who replies.