I'm somewhat confused about how to find this particular Fourier coefficient. Let me explain:(adsbygoogle = window.adsbygoogle || []).push({});

I have these two formulas from the book:

where [tex]T[/tex] is the period of the function [tex]f(x)[/tex].

[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):

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.

[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.

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# Homework Help: Fourier coefficient

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