How to check Fourier series solution (complex)

[PhysicsMark]

Homework Statement

Find the complex Fourier series for:

$$f(t)=t(1-t), 0<t<1$$

Homework Equations

$$f(t)=\sum_{n=-\infty}^{\infty}c_n{e^{i\omega_n{t}}}$$

$$c_n=\frac{1}{\tau}\int_{t_0}^{t_0+\tau}e^{-i\omega_n{t}}f(t)dt$$

$$\omega_n=2\pi{n}\quad\tau=1$$

The Attempt at a Solution

I solved for c_n. I want to check my answer. I can only think of checking it by graphing it out to a few (50 or so) terms. I tried to graph this in Maple with my value for c_n and it couldn't do it. After that, I tried to solve the entire problem in Maple and that also did not work.

I have a few more of these to do, and I'd like to make sure I am doing this correctly before I move on. Does anyone know how to check my value for the coefficient?

 

[gabbagabbahey]

Do your coefficients depend on your choice of $t_0$? Because the relevant equations you posted are not correct.

 

[PhysicsMark]

I found the coefficients, c_n, by integrating:

$$c_n=\int_{0}^{1}t(1-t)e^{-i2\pi{n}t}dt$$

Are you saying that this is not the correct method to find c_n?

 

[gabbagabbahey]

No, it isn't.

The Fourier coefficients are given by

$$c_n=\int_{-\frac{1}{2}}^{\frac{1}{2}}f(2\pi t)e^{-i\omega_n t}dt=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-int}dt$$

which is not equivalent to what you've posted. Where did you find that incorrect equation for the coefficients?

 

[PhysicsMark]

Thanks for pointing that out. I'll recalculate my coefficient value later today.

My tutorial lists the equation for c_n as:

$$c_n=\frac{1}{\lambda}\int_{x_0}^{x_0+\lambda}e^{-i{k_n}x}f(x)dx$$

In a paragraph above this equation, it states that:

"...But most applications involve either functions of position or of time. In the former case, the period of the function, $$\lambda$$, is more conventionally called the wavelength, and $$k_n=\frac{2\pi{n}}{\lambda}$$ is the wave number for the n'th mode. If time is the variable, however, the period is called, indeed, the period, and is usually represented by $$\tau$$. The n'th mode has the angular frequency, or often simply the frequency, $$\omega_n=\frac{2\pi{n}}{\tau}$$."

It did not explicitly give the equation for c_n in this case. I must have made a mistake converting from a spatial to time variable.

 

[vela]

No, it isn't.

The Fourier coefficients are given by

$$c_n=\int_{-\frac{1}{2}}^{\frac{1}{2}}f(2\pi t)e^{-i\omega_n t}dt=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-int}dt$$

which is not equivalent to what you've posted. Where did you find that incorrect equation for the coefficients?

Those expressions are for a function with period 2π. If you rescale for a function with period τ and allow for a shift (since it really doesn't matter which particular cycle you integrate over), you get PhysicsMark's integral.

 

[PhysicsMark]

Those expressions are for a function with period 2π. If you rescale for a function with period τ and allow for a shift (since it really doesn't matter which particular cycle you integrate over), you get PhysicsMark's integral.

Thanks for the clarification, Vela. I spoke with my professor today, and he also said the original integral should be correct.

He went on to say that I should be able to plot it in Maple. So, I guess that means I need some more practice in Maple (That should be no surprise to Vela...https://www.physicsforums.com/showthread.php?t=391887).

Thanks to Vela and Gabbagabbahey for replying.

 

[vela]

In case you want to check your answer, here's what I found for c_n for $n\ne 0$:

$$c_n=-\frac{1}{2n^2\pi^2}$$

When you plot the series, you should only need to use a handful of terms -- 50 would be overkill -- to see if it's summing to what you expect. Just remember to match the negative n's with positive n's so that the sum is real.

 

[PhysicsMark]

Thanks a lot! That is also the answer I got.

I realized why maple wouldn't plot it, it was because I did not account for the "n" in the denominator. No wonder Maple was blabbering about a singularity.