# Having trouble understanding Odd or even functions of Fourier

• Spoolx
In summary, the conversation discusses identifying whether a given function is even, odd, or neither. The concept of an even function is described as one where f(-t) = f(t) and an odd function as one where f(-t) = -f(t). The conversation also provides examples of even and odd functions and asks the question of what distinguishes an even function from an odd function on a graph. One of the examples given is a piecewise function and the conversation explains how to extend it to cover the entire domain in order to determine its symmetry. The conversation also includes two additional functions and discusses how to determine their symmetry.

## Homework Statement

Is the function even, odd, or neither

$$y(t) = \frac{2At}{w} for 0<t<\frac{w}{2}$$
$$y(t) = \frac{-2At}{w}+2A for \frac{w}{2}<t<w$$

## Homework Equations

even function f(-t) = f(t)
off function f(-t) = -f(t)

## The Attempt at a Solution

I just don't understand the concept, any help is appreciated.

An example of a even function is y = cosine (x) or y = x^2.
An example of an odd function is y = sine (x) or y = x.

If you look a graphs of these functions, what property of their graphs distinguishes an even function from an odd function?

I believe you meant to specify one piecewise function:
$$y(t) = \begin{cases} \frac{2At}{w}, & 0 \le t < \frac w2 \\ 2A - \frac{2At}{w}, & \frac w2 \le t < w \end{cases}.$$ This specifies just one period of the function. You want to repeat it so that you have a function defined for all ##t##. The question is asking if the resulting function is even, odd, or neither.

1 person
vela said:
I believe you meant to specify one piecewise function:
$$y(t) = \begin{cases} \frac{2At}{w}, & 0 \le t < \frac w2 \\ 2A - \frac{2At}{w}, & \frac w2 \le t < w \end{cases}.$$ This specifies just one period of the function. You want to repeat it so that you have a function defined for all ##t##. The question is asking if the resulting function is even, odd, or neither.

That is correct, I could not figure out how to do the piecewise function.

I have done a bunch of reading and I still don't get it. If i graph it (see attachment it doesn't appear to be symmetrical about either axis, or maybe I am just missing something)
Thanks

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Spoolx said:
That is correct, I could not figure out how to do the piecewise function.

I have done a bunch of reading and I still don't get it. If i graph it (see attachment it doesn't appear to be symmetrical about either axis, or maybe I am just missing something)
Thanks

What you have drawn is just one period. Draw some more of the graph, using the fact that you know what a period looks like, going both directions and see if what you get looks even or odd.

1 person
LCKurtz said:
What you have drawn is just one period. Draw some more of the graph, using the fact that you know what a period looks like, going both directions and see if what you get looks even or odd.

in that case I would say its symmetrical about the y-axis which I believe is like a cos function so even?

Spoolx said:
in that case I would say its symmetrical about the y-axis which I believe is like a cos function so even?

Yes. You can see from the graph that ##f(-x)=f(x)##.

1 person
Well I thought I understood it, but I ran into two more problems which don't make sense

To me, they both should be Cos or even functions but the book says the first one is an odd function.

What am I missing?

Thanks so much

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If u look at the functions, they both have a factor of t, so they are like sin(t) or just t, odd.

Spoolx said:
Well I thought I understood it, but I ran into two more problems which don't make sense

To me, they both should be Cos or even functions but the book says the first one is an odd function.

What am I missing?

Thanks so much

benorin said:
If u look at the functions, they both have a factor of t, so they are like sin(t) or just t, odd.

No benorin, that is not correct at all.

@Spoolx: Generally you should start a new thread with a new question. If you extend those functions by drawing more periods as you did before you will wind up with one of three possibilities: even, odd, or neither. Ignoring the book's answer, tell us what you think and why.

## 1. What are odd and even functions in Fourier analysis?

Odd and even functions are two types of functions commonly used in Fourier analysis. An odd function is a function where f(-x) = -f(x), meaning that it is symmetric about the origin. On the other hand, an even function is a function where f(-x) = f(x), meaning that it is symmetric about the y-axis.

## 2. How do odd and even functions behave in Fourier analysis?

In Fourier analysis, odd functions only have sine terms in their Fourier series, while even functions only have cosine terms. This is because the Fourier series for an odd function is multiplied by sine while the Fourier series for an even function is multiplied by cosine.

## 3. What are some examples of odd and even functions?

Examples of odd functions include sin(x), x^3, and tan(x), while examples of even functions include cos(x), x^2, and sec(x).

## 4. How can I determine if a function is odd or even using Fourier analysis?

To determine if a function is odd or even using Fourier analysis, you can use the symmetry properties of the function. If f(-x) = -f(x), then the function is odd. If f(-x) = f(x), then the function is even.

## 5. Why is it important to understand odd and even functions in Fourier analysis?

Understanding odd and even functions in Fourier analysis is important because it allows us to simplify the calculations involved in finding the Fourier series of a function. By identifying the type of function, we can determine which terms will be present in the Fourier series and make the analysis process more efficient.