- #1

Esran

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## Homework Statement

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].

## 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 [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.