# How to Find Fourier Series for a Given Function Using Sine Series?

• RJLiberator
In summary, the conversation discusses finding the Fourier series for a given function and the confusion surrounding the various equations and elements involved. The solution involves starting with the Fourier sine series and finding the coefficients by performing an integral. It is recommended to review the derivation of the formulas to better understand the concepts.
RJLiberator
Gold Member

## Homework Statement

Find the Fourier series for the following function (0 ≤ x ≤ L):
y(x) = Ax(L-x)

## The Attempt at a Solution

1. We start with the sum from n to infinity of A_n*sin(n*pi*x/L) where An = B_n*Ax(l-x)

2. We have the integral from 0 to L of f(x)*sin(m*pi*x/L) dx

I really have no idea what to do, I am francticlly looking through notes and websites. I understand the Fourier sine series should be pretty easy to find, it's just plugging in values, but there are so many different equations/elements.

Let me try this solution:

f(x) = L/pi(sum from n = 1 to infinity of sin(n*pi*x/L)

Ah?

When you are asked the for the Fourier series of a function, your answer should be the coefficients appearing in the sum.

RJLiberator
From my notes, would:

A_m = 2/L integral from 0 to L f(x)*sin(m*pi*x/L) dx be the answer then?

where f(x) = Ax(L-X)

Okay, let me say this:

sum from 1 to infinity of (b_n*sin(n*pi*x/L))

where b_n = 2/L integral from 0 to L (f(x)*sin(n*pi*x/L))

where f(x) = the function in question, namely Ax(L-x)

All together we have

The sum from n=1 to infinity of 2/L integral from 0 to L of (Ax(L-x))*sin^2(n*pi*x/L) dx

While formally correct, this doesn't answer the question. You have to find an expression for the b_n.

RJLiberator
b_n = 2/L integral from 0 to L (f(x)*sin(n*pi*x/L))
where f(x) = the function in question, namely Ax(L-x)

so b_n = 2/L integral from 0 to L (Ax(L-x)*sin(n*pi*x/L))
Is that incorrect for b_n?

RJLiberator said:
so b_n = 2/L integral from 0 to L (Ax(L-x)*sin(n*pi*x/L))
Is that incorrect for b_n?
It's a start. Now you have to perform the integral.

RJLiberator
RJLiberator said:
I really have no idea what to do, I am francticlly looking through notes and websites. I understand the Fourier sine series should be pretty easy to find, it's just plugging in values, but there are so many different equations/elements.
Now that you've been working with Fourier series for a while, it wouldn't hurt to go back and review the derivation of the various formulas (using one reference). If you understand the basics, all the variations/conventions will make more sense and won't seem so confusing.

RJLiberator

## 1. What is a Fourier sine series?

A Fourier sine series is a mathematical representation of a periodic function using a series of sine functions with different frequencies and amplitudes. It is used to decompose a complex function into simpler components, making it easier to analyze and understand.

## 2. How do you create a Fourier sine series?

To create a Fourier sine series, you need to follow a specific process called Fourier analysis. This involves finding the Fourier coefficients, which represent the amplitudes of the sine functions, by using integration techniques. These coefficients are then used to construct the Fourier sine series.

## 3. What is the purpose of creating a Fourier sine series?

The purpose of creating a Fourier sine series is to break down a complex function into simpler components. This allows for easier analysis and understanding of the function's behavior and properties. It is also used in various scientific fields, such as signal processing, to study and manipulate periodic signals.

## 4. What are the applications of Fourier sine series?

Fourier sine series has various applications in mathematics, physics, engineering, and other scientific fields. It is used in signal and image processing, data compression, solving differential equations, and solving boundary value problems. It is also used in studying the behavior of physical systems, such as vibrating strings and electrical circuits.

## 5. Are there any limitations to using Fourier sine series?

Although Fourier sine series is a powerful mathematical tool, it does have some limitations. It can only be used for functions that are periodic, continuous, and have finite energy. It also requires the function to have certain symmetries to simplify the calculation of Fourier coefficients. In some cases, other methods such as Fourier transform may be more suitable.

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