Fourier Series: To Factor or Not to Factor, That is the Question

[doive]

Homework Statement

I have conflicting notes and even textbooks about where i should and shouldn't have factors of 2 when doing Fourier series. I just want to check once and for all I'm doing it right and not introducing erroneous factors of 2 or omitting any

Homework Equations

$$\tilde{f}(x)= \frac{a_0}{2} + \sum^{\inf}_{n=1} [a_n cos(\frac{n \pi x}{L}) + b_n sin(\frac{n\pi x}{L})]$$

$$a_n = \frac{1}{L} \int^{L}_{-L} cos \frac{n\pi x}{L} f(x) dx$$

$$b_n = \frac{1}{L} \int^{L}_{-L} sin(\frac{n\pi x}{L}) f(x) dx$$

$$a_0 = \frac{1}{2L} \int^{L}_{-L} f(x) dx$$

where the interval is [-L,L] and the period is 2L

The Attempt at a Solution

Is the above formulation correct (i'm particularly unsure about the 1/2L associated with a₀)

 

[vela]

Just plug in the series for f(x) into the integrals and see if the coefficients work out correctly.

 

[jbunniii]

Is the above formulation correct (i'm particularly unsure about the 1/2L associated with a₀)

You have an extra factor of 1/2 associated with $a_0$. To see that this is the case, set $f(x) = 1$. Then according to your formulas, $a_0 = 1$ and the Fourier series is

$$\tilde{f}(x) = 1/2$$.

There should be a 1/2 either on the defining integral for $a_0$ or on the $a_0$ term in the series expansion, but not both.

You can check your other coefficients in a similar way by setting

$$f(x) = \cos(n\pi x/L)$$

or

$$f(x) = \sin(n \pi x/L)$$

 

[doive]

thank you guys!

I had tried checking it but couldn't be certain it wasn't my dodgy arithmetic!