# Why Does the Fourier Series of |sin(x)| Treat n=1 Differently?

• Kqwert
In summary, the person is having trouble finding the Fourier series of abs(sin(x)) and is questioning why the solution manual has separate calculations for n > 1 and n = 1. They also ask for clarification on what interval to use for the calculations.
Kqwert

## Homework Statement

Hello,

i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1 (i.e. a2, a3, a4, a5... and so on) and one for n = 1 (i.e. a1). Why?

## The Attempt at a Solution

Well, you need to compute ##\int cos(nx) sin(x) dx##. You can use a trig identity to rewrite that in terms of ##sin((n+1) x)## and ##sin((n-1)x)##. The case ##n=1## is special, because the second term is zero.

Kqwert
Thank you!

Kqwert said:

## Homework Statement

Hello,

i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1 (i.e. a2, a3, a4, a5... and so on) and one for n = 1 (i.e. a1). Why?

## The Attempt at a Solution

What interval are you using? ##[-\pi,\pi]?## ##[0, \pi]?## ##[0, 2\pi]?##

## What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It can be used to approximate any periodic function, including non-sinusoidal ones.

## What is the Fourier series of abs(sin(x))?

The Fourier series of abs(sin(x)) is an infinite sum of sine and cosine functions that approximates the absolute value of the sine function. It is given by the formula: f(x) = (4/pi) * sum(n=1 to infinity) (sin((2n-1)x)/(2n-1)).

## Why is the Fourier series of abs(sin(x)) important?

The Fourier series of abs(sin(x)) is important because it is a useful tool in analyzing and approximating periodic functions. It also has applications in fields such as signal processing, image analysis, and physics.

## How is the Fourier series of abs(sin(x)) calculated?

The Fourier series of abs(sin(x)) can be calculated using the formula mentioned above, which involves finding the coefficients of the sine and cosine terms. These coefficients can be determined using integration techniques or by using Fourier series tables.

## What are some real-world applications of the Fourier series of abs(sin(x))?

The Fourier series of abs(sin(x)) has applications in various fields such as audio and signal processing, image analysis, and physics. It is used to analyze and approximate periodic signals, filter out noise, and extract useful information from signals. It also plays a role in the study of vibrations and waves in physics.

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