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
$\displaystyle \sum_{n= 0}^\infty a_n sin(nx)+ b_n cos(nx)$
However, it is quite possible for most of those coefficients, $\displaystyle a_n$ and $\displaystyle b_n$, to be 0! In this case, you don't have to do any "calculation" at all- just compare
$\displaystyle \sum_{n= 0}^\infty a_n sin(nx)+ b_n cos(nx)$
and 20+ 10 sin(t).
Hi. I'm trying to learn Fourier Series (and the transform, too). This thread touches on a couple of questions I have, may I post them here?
The definition I have for the Fourier Series, $\displaystyle g(t)$ (with$\displaystyle P$ as the fundamental period), is:
$\displaystyle g(t)=\sum_{n=-\infty}^{\infty} c_n \cdot e^{i\frac{2 \pi n t}{P}}$
where $\displaystyle c_n$ is (using$\displaystyle v(t)$ as the original equation from the OP):
$\displaystyle c_n=\frac1T \int_0^T v(t) \cdot e^{-i\frac{2 \pi n t}{P}}dt$
I can use Euler's identity to get your definition, but you are starting your series at $\displaystyle 0$, the definition I have starts at $\displaystyle -\infty$. Why the indexing difference?
Using your definition, shouldn't $\displaystyle a_n=b_n$ for all $\displaystyle n$? 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 $\displaystyle P=\pi$ as the fundamental period). I tried comparing $\displaystyle v(t)=20+10sin(t)$ to your series definition. I get:
$\displaystyle a_1=10$
$\displaystyle a_n=0\; \forall n \ne 1$
$\displaystyle b_n=0\; \forall n$.
Is that right? If so, what is the final answer (the infinite Fourier Series)? What about the $\displaystyle 20$ in original equation?
I also tried the definition I have for $\displaystyle c_n$. 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.
HallsOfIvy posted the definition of a fourier series on real form. You have the complex form.
Complex Form of Fourier Series