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\begin_layout Section
Fourier Series and Complex Numbers
\end_layout

\begin_layout Standard
Fourier Analysis is usually taught using complex numbers rather than sines
 and cosines.
 While this has the disadvantage of requiring at least some familiarity
 of complex numbers, it makes notation much simpler, and is also conceptually
 more elegant.
 Furthermore, you will have to understand the complex-number notation in
 order to understand most references about Fourier Series.
 Therefore, we will quickly sketch the derivation of Fourier Series with
 complex numbers.
 Furthermore, this will set the stage for the Fourier Transform in the next
 section.
\end_layout

\begin_layout Itemize
A complex number 
\begin_inset Formula $z$
\end_inset

 consists of a real part and an imaginary part:
\begin_inset Formula 
\[
z=a+bi,\mbox{ where }i=\sqrt{-1}.
\]

\end_inset


\end_layout

\begin_layout Itemize
Any complex number 
\begin_inset Formula $z$
\end_inset

 can also be written in its 
\emph on
polar
\emph default
 form 
\emph on

\begin_inset Formula $z=Ae^{i\theta}$
\end_inset

, 
\emph default
where 
\begin_inset Formula $A$
\end_inset

 is the modulus of 
\begin_inset Formula $z$
\end_inset

, and 
\begin_inset Formula $\theta$
\end_inset

 is the argument of 
\begin_inset Formula $z$
\end_inset

.
 
\end_layout

\begin_layout Itemize
We can convert from the Cartesian form to the polar form by 
\begin_inset Formula $A=\sqrt{a^{2}+b^{2}},$
\end_inset

 
\begin_inset Formula $\theta=\tan^{-1}(\frac{b}{a}),$
\end_inset

 and from polar to Cartesian by 
\begin_inset Formula $a=A\sin(\theta),$
\end_inset

 
\begin_inset Formula $b=A\cos(\theta).$
\end_inset

 
\end_layout

\begin_layout Itemize
In other words, any complex number 
\begin_inset Formula $z$
\end_inset

 can be written as 
\begin_inset Formula $z=Ae^{i\theta}=A(cos\theta+i\sin\theta).$
\end_inset

This will allow us to collapse the sines and cosines into complex numbers.
\end_layout

\begin_layout Standard
The Fourier Series approximation of 
\begin_inset Formula $f(x)$
\end_inset

 of order 
\begin_inset Formula $N$
\end_inset

 in complex number notation is 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
f_{N}(x)=\sum_{k=-N}^{N}c_{k}e^{ikx}.
\]

\end_inset

 To find each coefficient 
\begin_inset Formula $c_{m},$
\end_inset

 we need to make use of the integration identity
\begin_inset Formula 
\[
\int_{-\pi}^{\pi}e^{ix(k-m)}dx=\begin{cases}
2\pi & \mbox{for }k=m\\
0 & \mbox{for }k\neq m.
\end{cases}
\]

\end_inset


\end_layout

\begin_layout Standard
We only need 
\emph on
one 
\emph default
such identity when we are using complex numbers, rather than several ones.
 Furthermore, this identity can quickly be derived by using the basic integratio
n rule 
\begin_inset Formula $\int e^{ax}dx=\frac{1}{a}e^{ax}.$
\end_inset

 Hence, each coefficient 
\begin_inset Formula $c_{m}$
\end_inset

 is found by
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\int_{-\pi}^{\pi}f(x)e^{-ixm}dx & = & \int_{-\pi}^{\pi}\left(\sum_{k=-\infty}^{\infty}c_{k}e^{ikx}\right)e^{-ixm}\\
 & = & \sum_{k=-\infty,k\neq m}^{\infty}\int_{-\pi}^{\pi}c_{k}e^{ix(k-m)}dx+\int_{-\pi}^{\pi}c_{m}e^{ix(m-m)}dx\\
 & = & 0+2\pi c_{m}\\
\implies c_{m} & = & \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-ixm}dx.
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
It should be noted that, while each coefficient 
\begin_inset Formula $c_{m}$
\end_inset

 could be an imaginary number, the Fourier approximation 
\begin_inset Formula $f_{N}(x)$
\end_inset

 is actually a real-valued function, as the imaginary parts of its components
 magically cancel out.
 
\end_layout

\begin_layout Standard

\end_layout

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