Is This Periodic Function Even, Odd, or Neither?

[liam2708]

Hello I am stuck on the following question:

Consider the following graph of a periodic function, period T.
http://img157.imageshack.us/img157/4337/waveme4.png

(a) Clearly giving the reasons, state whether this function is even, odd or neither.

(b) With reference to all relevant symmetry features, state which parts of which harmonics are present in the Fourier series representation of this periodic function.

I have attempted part A and think I have done it correctly, however I have no idea where to start with part b. Any help would be greatly appreciated.

Thanks!

 

[Hurkyl]

(a) Clearly giving the reasons, state whether this function is even, odd or neither.

(b) With reference to all relevant symmetry features, state which parts of which harmonics are present in the Fourier series representation of this periodic function.

Doesn't the answer to part (a) tell you things about the Fourier series?

 

[piercas]

Hello!

First of all, great job on part A! From the given graph, we can see that the function is symmetric about the y-axis (even). This is because for every point (x,y) on the graph, there is also a corresponding point (-x,y). This means that the function remains unchanged when we replace x with -x.

Now, onto part B. In order to determine the harmonics present in the Fourier series representation of this periodic function, we need to look at its symmetry features. The function is symmetric about the y-axis, which means that only cosine terms will be present in the Fourier series. This is because cosine is an even function, meaning that cos(-x) = cos(x).

Additionally, we can see that the function is also symmetric about the x-axis. This means that only even harmonics will be present in the Fourier series. This is because the cosine function is also symmetric about the x-axis, so only even multiples of the fundamental frequency will contribute to the function.

Therefore, the Fourier series representation of this periodic function will only contain even multiples of the fundamental frequency (cosine terms). This includes the fundamental frequency itself (cos(1x)), as well as the second harmonic (cos(2x)), fourth harmonic (cos(4x)), and so on.

I hope this helps! Remember to always consider the symmetry features of a function when determining its Fourier series representation. Keep up the good work!