# Fourier Series and the first term

• rdfloyd
In summary: He also said that the 1/2 is there to make sure that the integrand is in fact an integrand and not a function of x that is just a cosine.
rdfloyd
I wasn't really sure where to post this because I am covering this in 2 classes (Math and Physics). Figured this would be my best bet.

The Fourier series of some Function is $a_{0}/2+etc...$. I've looked in several textbooks but none explain why the $1/2$ is there, and not in any of the other terms of the summation.

I do have a homework problem concerning this, but my professor said it's ok to not explain this part of the Fourier series. I'm intrigued by this now, so I'd like to know.

Remember that the general term of a Fourier series is

$$<f,e_i>$$

where the $e_i$ is normalized (has norm 1)

We want to define

$$a_n=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)dx$$

If n>0, then this holds. The $1/\pi$ comes from normalizing the cosine. That is, the above is actually equal to

$$<f,\cos nx>$$

but cos(nx) does not have norm 1, but rather pi. So we must divide by pi to normalize.

We want the formula for an to hold for n=0 as well. But in this case, we have

$$<f,1>$$

and the 1 is not normalized and has norm 2pi. So in order to normalize the thing, we must divide by 2pi. Division by pi is already taken care of in the definition of an, so we must also divide by 2.

So, why doesn't the 2 tag along with the rest of the terms? In this case, it seems like there is a piecewise function under conditions n=0 and n>0.

Sorry if I'm asking a dumb question.

Integrating from -pi to pi:
The function f=1 yields 2pi
The function f=cos2(nx) yields pi.

Last edited:
Office_Shredder said:
Integrating from -pi to pi:
The function f=1 yields 2pi
The function f=cos(nx) yields pi.
You should have cos2(nx), not cos(nx).

mathman said:
You should have cos2(nx), not cos(nx).

Thanks. I edited my original post to avoid confusing anybody

I watched a video by MIT's OCW for 18.03 when I was doing this a while back, and the prof derived the Fourier series and explained the 1/2 quite well

## 1. What is a Fourier Series?

A Fourier Series is a mathematical representation of a periodic function as a sum of sinusoidal functions. It is used to analyze and approximate any periodic function into a combination of simpler functions.

## 2. What is the purpose of the first term in a Fourier Series?

The first term in a Fourier Series is also known as the DC or constant term. It represents the average value of the function over one period and is used to shift the entire series up or down.

## 3. How is the first term calculated in a Fourier Series?

The first term in a Fourier Series is calculated by finding the average value of the function over one period. This is done by taking the integral of the function over one period and dividing it by the length of the period.

## 4. Why is the first term important in Fourier Series analysis?

The first term in a Fourier Series is important because it allows us to shift the entire series up or down, making it easier to analyze and compare different functions. It also helps in approximating the original function with a finite number of terms.

## 5. Can the first term be zero in a Fourier Series?

Yes, the first term in a Fourier Series can be zero if the function is symmetric about the x-axis. In this case, the average value of the function over one period would be zero, resulting in a first term of zero.

Replies
11
Views
1K
Replies
8
Views
4K
Replies
1
Views
1K
Replies
3
Views
849
Replies
3
Views
2K
Replies
3
Views
8K
Replies
1
Views
567
Replies
6
Views
1K
Replies
8
Views
3K
Replies
5
Views
3K