%%%%		-*- Mode: LaTeX -*-
%%%% Author:    J. Robert Buchanan
%%%% Address:   Department of Mathematics
%%%%		Millersville University
%%%%            P.O. Box 1002
%%%%            Millersville, PA 17551-0302
%%%% Phone:     717-871-7305
%%%% FAX:	717-871-7948
%%%%
%%%% Email:     Robert.Buchanan@millersville.edu
%%%%
%%%% Date:      02/12/2025
%%%%
%%%% Purpose:   Complex Form of Fourier Series 
%%%%

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\title{Complex Form of Fourier Series}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
% - Use the \inst{?} command only if the authors have different
%   affiliation.

\institute[Millersville University of Pennsylvania]
% (optional, but mostly needed)
{
  Department of Mathematics
}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date{ % (optional)
Fall
%Spring
%Summer
\the\year}

\subject{Talks}
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% out. 



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% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSubsection[]
{
  \begin{frame}<beamer>
    \frametitle{Outline}
    \tableofcontents[currentsection,currentsubsection]
  \end{frame}
}


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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
  % You might wish to add the option [pausesections]
%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%\section{Introduction}

\begin{frame}\frametitle{Objectives}

In this lesson we will learn:
\begin{itemize}
\item the Euler identity,
\item the complex function representation of Fourier series, and
\item review the hyperbolic trigonometric functions.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Euler Identity}

Let $i = \sqrt{-1}$, then for any real number $\theta$,
\[
e^{i \theta}
	= \cos \theta + i \sin \theta	.
\]
\textbf{Remark}:
this provides a connection between the exponential function and the
ordinary trigonometric functions.

\end{frame}
%%

\begin{frame}\frametitle{Related Identities}

\begin{align*}
\cos \theta
	&= \frac{e^{i \theta} + e^{-i \theta}}{2}	\\
\sin \theta
	&= \frac{e^{i \theta} - e^{-i \theta}}{2i}	.
\end{align*}

\uncover<2->{%
Note how these identities mirror the definitions of the hyperbolic
trigonometric functions.
\begin{align*}
\cosh \theta
	&= \frac{e^{\theta} + e^{-\theta}}{2}	\\
\sinh \theta
	&= \frac{e^{\theta} - e^{-\theta}}{2}	.
\end{align*}
}

\end{frame}
%%

\begin{frame}\frametitle{Transition to Complex Form (1 of 2)}

Suppose $f(x)$ is integrable on $[-L, L]$.
\begin{align*}
f(x)
	&\sim \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[ a_{n} \cos
  \left( \frac{n \pi x}{L} \right) + b_{n} \sin \left( \frac{n \pi
  x}{L} \right) \right]	\\
\uncover<2->{%
	&= \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[
   \frac{a_{n}}{2} \left( e^{\frac{i n \pi x}{L}} + e^{\frac{-i n \pi
   x}{L}} \right) + \frac{b_{n}}{2 i} \left( e^{\frac{i n \pi x}{L}}
   - e^{\frac{-i n \pi x}{L}} \right) \right]	\\
\uncover<3->{%
	&= \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[
   \frac{a_{n}}{2} \left( e^{\frac{i n \pi x}{L}} + e^{\frac{-i n \pi
   x}{L}} \right) - \frac{i b_{n}}{2} \left( e^{\frac{i n \pi x}{L}}
   - e^{\frac{-i n \pi x}{L}} \right) \right]	\\
	&= \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[
   \frac{a_{n} - i b_{n}}{2} e^{\frac{i n \pi x}{L}} + \frac{a_{n} + i
   b_{n}}{2} e^{\frac{-i n \pi x}{L}} \right]
}
}
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Transition to Complex Form (2 of 2)}

Suppose $f(x)$ is integrable on $[-L, L]$.
\[
f(x)
	\sim \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[
   \frac{a_{n} - i b_{n}}{2} e^{\frac{i n \pi x}{L}} + \frac{a_{n} + i
   b_{n}}{2} e^{\frac{-i n \pi x}{L}} \right]
\]
Define $b_{0} = 0$, $c_{0} = \frac{a_{0} + i b_{0}}{2}$, $c_{n} =
\frac{a_{n} - i b_{n}}{2}$, and $c_{-n} = \frac{a_{0} + i b_{n}}{2}$, then
\begin{align*}
f(x)
	&\sim c_{0} + \sum_{n=1}^{\infty} \left[ c_{n} e^{\frac{i n
   \pi x}{L}} + c_{-n} e^{\frac{-i n \pi x}{L}} \right]	\\
	= \sum_{n=-\infty}^{\infty} c_{n} e^{\frac{i n \pi x}{L}}	.
\end{align*}
This is the Fourier series for $f(x)$ in complex form.

\end{frame}
%%

\begin{frame}\frametitle{Complex Coefficients (1 of 2)}

For $n = 0, 1, 2, \ldots$,
\begin{align*}
c_{n}
	&= \frac{a_{n} - i b_{n}}{2}
	= \frac{1}{2} \left(
\frac{1}{L} \int_{-L}^{L} f(x) \cos \frac{n \pi x}{L} \, dx -
\frac{i}{L} \int_{-L}^{L} f(x) \sin \frac{n \pi x}{L} \, dx
\right)	\\
	&= \frac{1}{2L} \int_{-L}^{L} f(x) \left(
\cos\frac{n \pi x}{L} - i \sin \frac{n \pi x}{L}
\right) \, dx	\\
	&= \frac{1}{2L} \int_{-L}^{L} f(x) e^{-i n \pi x/L} \, dx	.
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Complex Coefficients (2 of 2)}

For $n = 1, 2, \ldots$,
\begin{align*}
c_{-n}
	&= \frac{a_{n} + i b_{n}}{2}
	= \frac{1}{2} \left(
\frac{1}{L} \int_{-L}^{L} f(x) \cos \frac{n \pi x}{L} \, dx +
\frac{i}{L} \int_{-L}^{L} f(x) \sin \frac{n \pi x}{L} \, dx \right)	\\
	&= \frac{1}{2} \left(
\frac{1}{L} \int_{-L}^{L} f(x) \cos \frac{-n \pi x}{L} \, dx -
\frac{i}{L} \int_{-L}^{L} f(x) \sin \frac{-n \pi x}{L} \, dx
\right)	\\
	&= \frac{1}{2L} \int_{-L}^{L} f(x) e^{i n \pi x/L} \, dx	.
\end{align*}
\uncover<2->{%
Therefore for any $n \in \mathbb{Z}$,
\[
c_{n}
	= \frac{1}{2 L} \int_{-L}^{L} f(x) e^{-i n \pi x/L} \, dx	.
\]
}

\end{frame}
%%

\begin{frame}\frametitle{Example}

Suppose $i = \sqrt{-1}$ and $a$ is a constant with $a \not\in \{ k \,
i \, | \, k = 0, 1, 2, \ldots \}$.
Find the complex form of the Fourier series of $f(x) = e^{a x}$ on
$(-\pi, \pi)$.

\end{frame}
%%

\begin{frame}<handout:0>\frametitle{Solution}

Using the formula for the complex Fourier coefficients,
\begin{align*}
c_{n}
	&= \frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{a x} e^{-i n x} \, dx	\\
	&= \frac{1}{2 (a - i n) \pi} \left(
e^{(a - i n) \pi} - e^{-(a - i n) \pi} \right)	\\
	&= \frac{1}{(a - i n) \pi} \left(
\frac{e^{(a - i n) \pi} - e^{- (a  - i n) \pi}}{2} \right)	\\
	&= \frac{\sinh (a - i n) \pi}{(a - i n) \pi}	.
\end{align*}
\uncover<2->{%
Therefore the complex form of the Fourier series is
\[
e^{a x}
	\sim \sum_{n=-\infty}^{\infty} \frac{\sinh (a - i n) \pi}{(a -
          i n) \pi} e^{i n x}	.
\]
}

\end{frame}
%%

%\begin{frame}\frametitle{Homework}

%\begin{itemize}
%\item Read Section $x.y$
%\item Exercises: 
%\end{itemize}

%\end{frame}
%%

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