Alternate expressions of Fourier series formula

[Esran]

Homework Statement

Show that the Fourier series formula $$F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt))$$ can be expressed as $$F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}c_{n}cos(nwt-\phi_{n})$$. Relate the coefficients $$c_{n}$$ to $$a_{n}$$ and $$b_{n}$$.

Homework Equations

We have the usual equations for the coefficients of a Fourier series.

The Attempt at a Solution

I'm really just checking the integrity of my solution here. I want to be sure I did not misunderstand the nature of $$\phi_{n}$$ or anything else.

Let $$n\in Z^{+}$$. Pick $$\phi_{n}$$ such that $$\frac{a_{n}}{cos(\phi_{n})}=\frac{b_{n}}{sin(\phi_{n})}$$. We know we can do this since we could just choose $$\phi_{n}=nw_{0}t$$, where $$w_{0}=2 \pi f$$, the fundamental frequency of the Fourier series. Let $$c_{n}=\frac{a_{n}}{cos(\phi_{n})}=\frac{b_{n}}{sin(\phi_{n})}$$.

$$F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt))$$
$$F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(c_{n}cos(\phi_{n})cos(nwt)+c_{n}sin(\phi_{n})sin(nwt))$$
$$F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(c_{n}cos(nwt-\phi_{n}))$$

Which completes the problem.

 

[diazona]

Your proof seems fine, but the one thing I would point out is that φ_n is supposed to be a constant. So you shouldn't pick φ_n = nω₀t, because that would make the phase constant a linear function of time. Instead, just solve the equation
$$\frac{a_n}{\cos\phi_n} = \frac{b_n}{\sin\phi_n}$$
for φ_n, getting
$$\phi_n = \tan^{-1}\frac{b_n}{a_n}$$
(I'll leave the investigation of the case a_n=0 to you ) Using that value of φ_n, you can define c_n the same way you did, and the rest of the proof should be unchanged.

 

[Esran]

Thanks!