- #1

TheSodesa

- 224

- 7

## Homework Statement

The function ##f## is defined as follows:

\begin{equation*}

f(t) =

\begin{cases}

1, \text{ when } 2k < t < (2k+1),\\

0, \text{ when } t = k,\\

2, \text{ when } (2k-1) < t < 2k, & k \in \mathbb{Z}\\

\end{cases}

\end{equation*}

What is the period ##T## of the function ##f##? How about angular frequency? Calculate the following definite integrals:

[tex]

a_n = \frac{2}{T}\int_{0}^{T} f(t) \cos (n \omega t) dt

\text{ and }

b_n = \frac{2}{T}\int_{0}^{T} f(t) \sin (n \omega t) dt,

[/tex]

where ##n = 0,1,2, \ldots##

## Homework Equations

Angular freguency:

\begin{equation}

\omega = 2 \pi f = \frac{2 \pi}{T}

\end{equation}

## The Attempt at a Solution

I should preface this by saying, that something in me thinks this should be a simple problem, and that I'm just confused by the notation used.

Anyways, I started by cataloging how the function should behave with different values of ##k \in \mathbb{Z}##:

\begin{array}{ | l | c | c | c | c | c | c |}

\hline

k & 2k-1 & 2k & 2k+1 & f(t), (2k-1) < t < 2k & f(t), t = k & f(t), 2k < t < (2k+1)\\

\hline

0 & -1 & 0 & 1 & 2 & 0 & 1\\

1 & 1 & 2 & 3 & 2 & 0 & 1\\

2 & 3 & 4 & 5 & 2 & 0 & 1\\

3 & 5 & 6 & 7 & 2 & 0 & 1\\

4 & 7 & 8 & 9 & 2 & 0 & 1\\

\hline

\end{array}

It looks like ##f## will repeat in intervals of ##2##, since whenever ##k## is increased by ##1##, the domain of ##f## is shifted by ##2## to the right, and within that new domain ##f## is defined identically. Now here's the first thing that confuses me: I know I can test for periodicity by plugging ##t+T## in place of ##t## in ##f(t)##. However, here it wouldn't create a nice expression that simplified into the original through trig symmetry or some other trick. How can I show in writing, that what I stated above is true, or is the above intuitive realization enough for a proof?

Moving on, assuming the period was 2 implies that ##\omega = \frac{2 \pi}{2} = \pi##. Which leads us into the integration part of the problem, where another confusing thing occurs. Evaluating the integral seems simple enough: just split it into two parts between ##0## and ##T = 2## based on how ##f## is defined. However, in the assignment it is stated that ##n \in {0} \cup \mathbb{N}##. Does this mean I need to evaluate the integral infinitely many times, since the expression contains an ##n## that gains all of those values? Of course not, but I have no idea what this means in practise.

Help a derp, please?