# Finding the Fourier Series: A Beginner's Guide

• briteliner
In summary, the conversation was about finding the Fourier series for a piecewise function with bounds that were initially incorrect. The problem was to find the Fourier series for the function f(x)=sin(x) on the interval [-pi,pi], where f(x)=0 for x in [-pi,0] and f(x)=sin(x) for x in [0,pi]. The solution involved integrating sin(x)*sin(nx) and cos(nx) using trig product to sum formulae.
briteliner

## Homework Statement

find the Fourier series for the function

0 -pi<x<pi
f(x)={
sinx 0<x<pi

## The Attempt at a Solution

I don't know how to do this and can't find a good explanation anywhere, any kind of help is appreciated...even just a push in the right direction

Last edited:
Are you saying your function is
f(x) = 0 x<-pi
= sin(x) 0<x<pi
= 0 x>pi
If so, that is not periodic and does not have a Fourier series in the usual sense, although it probably has a Fourier transform. Is that what you want?

we are supposed to assume that the function is continued outside of the interval with period 2pi periodically.
i just realized there was a mistake with the bounds also, but i fixed it now. it is a piecewise function which equals 0 from -pi to pi and sinx from 0 to pi

I'm going to guess that the problem is to find the Fourier series on the interval [-pi,pi] of the function f(x)=sin(x) for x in [0,pi] and f(x)=0 for x in [-pi,0]. If so, then the coefficients of that series are defined in terms of integrals of sin(x) times sin(nx) and cos(nx). Since f(x)=0 for x<0, just integrate from 0 to pi. Can you do any of them?

ok thanks so then i have integral from 0 to pi of sinxcosnx dx. would this just give 0 since sinx is odd and cosnx is even, giving an odd function?

oh right. thanks so much!

## 1. What is the Fourier Series?

The Fourier Series is a mathematical tool used to represent a periodic function as a sum of simpler trigonometric functions. It is named after French mathematician Joseph Fourier and is widely used in signal processing, physics, and engineering.

## 2. Why is the Fourier Series important?

The Fourier Series is important because it allows us to analyze and understand complex periodic functions in terms of simpler components. This can be applied to a wide range of fields, including sound and image processing, vibration analysis, and data compression.

## 3. How do you find the Fourier Series of a function?

To find the Fourier Series of a function, you need to follow a series of steps including determining the period of the function, calculating the coefficients using integration, and representing the function as a sum of the coefficients and trigonometric functions. There are also various techniques and formulas that can be used to make the process easier.

## 4. What are some applications of the Fourier Series?

The Fourier Series has numerous applications in different fields, such as image and sound processing, telecommunications, control systems, and quantum mechanics. It is also used in solving differential equations and in understanding the behavior of physical systems.

## 5. Is there any software or tools available for finding the Fourier Series?

Yes, there are many software and tools available for finding the Fourier Series, such as MATLAB, Mathematica, and Wolfram Alpha. These tools can perform the necessary calculations and provide graphical representations of the series, making it easier for beginners to understand and use the concept.

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