(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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 [tex]\phi_{n}[/tex] or anything else.

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

[tex]F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt))[/tex]

[tex]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))[/tex]

[tex]F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(c_{n}cos(nwt-\phi_{n}))[/tex]

Which completes the problem.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Alternate expressions of Fourier series formula

**Physics Forums | Science Articles, Homework Help, Discussion**