- #1
Zaare
- 54
- 0
I'm somewhat confused about how to find this particular Fourier coefficient. Let me explain:
I have these two formulas from the book:
[tex]
(1) \qquad f\left(x\right) = \frac{a_0}{2}+\sum_{n=1}^\infty\left[a_n cos\left(n \Omega x\right)+b_n sin\left(n \Omega x \right)\right],
\quad \Omega = \frac{2 \pi}{T}
[/tex]
[tex]
(2) \qquad b_n = \frac{2}{T} \int_{a}^{a+T} f(x) sin(n \Omega x) dx
[/tex]
where [tex]T[/tex] is the period of the function [tex]f(x)[/tex].
And I'm supposed to find the Fourier coeffecient in the following:
[tex]
(3) \qquad f(x) = 2 \delta (x-1) + \delta (x-2) = \sum_{n=1}^\infty b_n sin\left(\frac{n \pi x}{3} \right), \quad 0 \leq x \leq 3.
[/tex]
Now, if I compare (3) and (1):
[tex]
sin\left(\frac{n \pi x}{3} \right) = sin(n \Omega x) \Rightarrow \Omega = \frac{\pi}{3} \Rightarrow T = 6.
[/tex]
According to (2), I'm supposed to use 0 and 6 as the limits for the integral. But (3) limits x to between 0 and 3.
So what limits am I supposed to use for the integral, and why?
I'd appreciate any help.
I have these two formulas from the book:
[tex]
(1) \qquad f\left(x\right) = \frac{a_0}{2}+\sum_{n=1}^\infty\left[a_n cos\left(n \Omega x\right)+b_n sin\left(n \Omega x \right)\right],
\quad \Omega = \frac{2 \pi}{T}
[/tex]
[tex]
(2) \qquad b_n = \frac{2}{T} \int_{a}^{a+T} f(x) sin(n \Omega x) dx
[/tex]
And I'm supposed to find the Fourier coeffecient in the following:
[tex]
(3) \qquad f(x) = 2 \delta (x-1) + \delta (x-2) = \sum_{n=1}^\infty b_n sin\left(\frac{n \pi x}{3} \right), \quad 0 \leq x \leq 3.
[/tex]
Now, if I compare (3) and (1):
[tex]
sin\left(\frac{n \pi x}{3} \right) = sin(n \Omega x) \Rightarrow \Omega = \frac{\pi}{3} \Rightarrow T = 6.
[/tex]
So what limits am I supposed to use for the integral, and why?
I'd appreciate any help.