Square wave exponential fourier series

[Funnynick]

This is A and B my friend is telling me that Co is actually 0 and I am getting 1/2 and i don't see exactly what I am doing wrong if i indeed am doing something wrong hopefully someone here can check this out and let me know exactly where i went wrong..
Thanks

 

[DrClaude]

You wrote yourself that $c_n = 0$ when $n$ is even.

 

[rude man]

I can't get the second image to open.

 

[Funnynick]

Thanks for the reply, i figured out my mistake and redid the problem completely and found the correct answer.

Thanks

 

[rezeew]

for sharing your question with me. It's great to see that you are exploring the concepts of Fourier series, which are an important tool in analyzing periodic functions.

To address your specific question, I would like to clarify a few things about the Square wave exponential Fourier series. First, it is important to understand that the Fourier series represents a periodic function as a sum of sinusoidal functions with different frequencies and amplitudes. The coefficients A and B in this case represent the amplitudes of the cosine and sine terms, respectively.

Now, to find the coefficients of the Fourier series, we use the complex exponential form of the Fourier series. This means that the coefficients A and B can be calculated using the following formulas:

A = 1/T * ∫f(t)cos(nωt)dt
B = 1/T * ∫f(t)sin(nωt)dt

Where T is the period of the function, ω is the fundamental frequency (2π/T), and n is the harmonic number.

In the case of a square wave, the function is only defined for half of the period and is equal to 1 for half of the period and equal to -1 for the other half. This means that the integral will be non-zero only for certain values of n. For n = 0, the integral will be 0 since the function is equal to 0 for half of the period. This is why the coefficient C0, which represents the average value of the function, is equal to 0.

For n = 1, the integral will be equal to 1 for half of the period and -1 for the other half. This means that the coefficient C1, which represents the amplitude of the first harmonic, will be equal to 1. Similarly, for n = 3, the integral will be equal to 1/3 for half of the period and -1/3 for the other half, resulting in a coefficient of 1/3 for the third harmonic.

I hope this explanation helps to clarify the concept of the Square wave exponential Fourier series and why the coefficient C0 is equal to 0. Keep exploring and experimenting with Fourier series, and you will gain a deeper understanding of this powerful tool in signal analysis.