How to find the period of a Fourier Series

[thecaptain90]

Hi. My problem is that even though I can find the Fourier series, its coefficients etc. I have trouble determining the period of equation. For example let's say we have f(t) = t, t $\in$ [-pi,pi]. I thought that the period was 2pi but in the solution it says that the period is pi. This isn't the first time that I couldn't determine the period of an equation in order to find the Fourier series. Can someone make things clear for me?
Thanks.

 

[HallsofIvy]

Are all the odd coefficients 0? If that is true, then the function is an even function, f(-x)= f(x), so that its values from $-\pi$ to 0 are identical to its values from 0 to [itex]\pi. That, together with the obvioius "periodic with period $2\pi$" means it is periodic with period $\pi$.

 

[mathman]

Are all the odd coefficients 0? If that is true, then the function is an even function, f(-x)= f(x), so that its values from $-\pi$ to 0 are identical to its values from 0 to [itex]\pi. That, together with the obvioius "periodic with period $2\pi$" means it is periodic with period $\pi$.

Not so: The function from -π to 0 is the mirror image of the function from 0 to π. In order to have a period of π, it needs to be a translate in this domain.

 

[LCKurtz]

Hi. My problem is that even though I can find the Fourier series, its coefficients etc. I have trouble determining the period of equation. For example let's say we have f(t) = t, t $\in$ [-pi,pi]. I thought that the period was 2pi but in the solution it says that the period is pi. This isn't the first time that I couldn't determine the period of an equation in order to find the Fourier series. Can someone make things clear for me?
Thanks.

The statement of the problem needs to give you enough information to determine the period. In your example, if the FS it to represent the function on (-pi,pi), its period must be at least 2pi. But unless you state that the given formula represents one period, you can't say for sure what FS you are talking about. For example, if you take g(t) = t on (-2pi,2pi) and write a 4pi periodic FS for it, it will equal your f(t) on (-pi,pi).