# Fourier Series for Periodic Functions - Self Study Problem

• Gopal Mailpalli
In summary, the conversation discusses the topic of self study and provides a homework statement about a periodic function with periodicity 2π. The equations for A, B, and C coefficients are given, along with an attempt at a solution. However, the formula for B coefficients is incorrect as it does not account for even index values. The user is advised to type the text or provide a clearer picture for better understanding.
Gopal Mailpalli
Self Study
1. Homework Statement

Consider a periodic function f (x), with periodicity 2π,

## Homework Equations

##A_{0} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)dx##
##A_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)cos\frac{2\pi rx}{L}dx##
##B_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)sin\frac{2\pi rx}{L}dx##

## The Attempt at a Solution

##A_{0} = C##
##A_{n} = 0##
##B_{n} = \frac{-C}{\pi r}cos\pi r##
http://imgur.com/a/4Q2oL

Your
Gopal Mailpalli said:
Self Study
1. Homework Statement

Consider a periodic function f (x), with periodicity 2π,

## Homework Equations

##A_{0} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)dx##
##A_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)cos\frac{2\pi rx}{L}dx##
##B_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)sin\frac{2\pi rx}{L}dx##

## The Attempt at a Solution

##A_{0} = C##
##A_{n} = 0##
##B_{n} = \frac{-C}{\pi r}cos\pi r##
http://imgur.com/a/4Q2oL
This post should have been sent to Calculus and Beyond Forum.

Your picture is hardly readable. Better to type the text in, or write it clearly and make a better picture.
The formula for the b coefficients is not correct. What happens in case of even index? What is b2, for example?

## 1) What is a Fourier series problem?

A Fourier series problem is a mathematical question that involves finding the coefficients of a Fourier series for a specified function. A Fourier series is a way to represent a periodic function as a sum of trigonometric functions.

## 2) What is the purpose of solving a Fourier series problem?

The purpose of solving a Fourier series problem is to accurately represent a periodic function as a sum of simpler trigonometric functions. This can be useful in analyzing and understanding the behavior of a function, as well as in practical applications such as signal processing and image compression.

## 3) How do you solve a Fourier series problem?

To solve a Fourier series problem, you first need to determine the period of the function. Then, you can use the formulas for the coefficients of the Fourier series to find the values of each coefficient. These formulas involve integrals and may require some algebraic manipulation. Once you have the coefficients, you can write out the Fourier series for the given function.

## 4) What are some applications of Fourier series?

Fourier series have many applications in mathematics, physics, and engineering. They are commonly used in signal processing to analyze and manipulate signals, in electrical engineering to design filters and amplifiers, and in quantum mechanics to describe the behavior of waves. They are also used in image and audio compression algorithms.

## 5) Are there any limitations to Fourier series?

While Fourier series are a powerful tool for representing periodic functions, they do have some limitations. They can only accurately represent functions that are periodic and have a finite number of discontinuities. They also may not converge for all functions, and the convergence may not be uniform. Additionally, the Fourier series may not accurately capture the behavior of a function near sharp corners or edges.

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