# Is there any symmetry I can use to find this Fourier sine series?

• richyw
In summary, the conversation involves finding the Fourier Sine Series of the function f(x)=x(\pi^2-x^2) using the given equations and attempting to solve it by using integration by parts. It is suggested to look for symmetry in the function and it is noted that the function is odd, allowing for the use of the half range sine expansion. It is also advised to do the integration by parts all at once rather than in separate steps.
richyw

## Homework Statement

I am going over a practice exam, and I need to find the FSS of $$f(x)=x(\pi^2-x^2)$$

## Homework Equations

$$f(x) \sim \sum^\infty_{n=1}a_n sin\left(\frac{n \pi x}{L}\right)$$
$$a_n=\frac{2}{L}\int^L_0 f(x)sin\left(\frac{n\pi x}{L}\right)dx$$

## The Attempt at a Solution

I think I need to integrate$$\frac{2}{\pi}\int^\pi_0 x(\pi^2-x^2)sin(n\pi)dx$$which is two integrals, the first one would need me to use IBP once, and the second one would need me to use IBP three times. This is on a practice exam (and my exam is in an hour), so I am guessing that this integral is easier if I can find some symmetry in it. Is this true?

You can indeed use symmetry. Graph f(x) on your calc: what do you notice? remember if x(t)=x(-t) then there is even symmetry, x(t)=-x(-t) means it has odd symmetry.

richyw said:

## Homework Statement

I am going over a practice exam, and I need to find the FSS of $$f(x)=x(\pi^2-x^2)$$

## Homework Equations

$$f(x) \sim \sum^\infty_{n=1}a_n sin\left(\frac{n \pi x}{L}\right)$$
$$a_n=\frac{2}{L}\int^L_0 f(x)sin\left(\frac{n\pi x}{L}\right)dx$$

## The Attempt at a Solution

I think I need to integrate$$\frac{2}{\pi}\int^\pi_0 x(\pi^2-x^2)sin(n\pi)dx$$which is two integrals, the first one would need me to use IBP once, and the second one would need me to use IBP three times. This is on a practice exam (and my exam is in an hour), so I am guessing that this integral is easier if I can find some symmetry in it. Is this true?

The fact that the function is odd is the reason you can use the half range sine expansion in the first place. That is why there are no cosine terms. You just need to bite the bullet and do the integration by parts. Don't do two separate integrals though. Write your integrand as$$(\pi^2x-x^3)sin(n\color{red}x)$$and do it all at once. Note your typo correction.

## 1. What is a Fourier sine series?

A Fourier sine series is a mathematical representation of a periodic function using only sine functions. It is a form of Fourier series, which is a way to decompose a function into simpler trigonometric functions.

## 2. How is symmetry used to find a Fourier sine series?

Symmetry is used to simplify the process of finding a Fourier sine series. If a function has certain symmetries, such as being odd or even, the coefficients in the series can be determined without having to do complicated integrals or calculations.

## 3. What types of symmetries can be used to find a Fourier sine series?

The most commonly used symmetries are odd symmetry, even symmetry, and half-wave symmetry. Odd symmetry means that the function is symmetric about the origin, while even symmetry means that the function is symmetric about the y-axis. Half-wave symmetry means that the function is symmetric about the midpoint of the period.

## 4. Can symmetry always be used to find a Fourier sine series?

No, not all functions have symmetries that can be used to simplify the calculation of a Fourier sine series. Some functions may have a combination of symmetries or may not have any symmetries at all. In these cases, other methods, such as integration by parts, may be used to find the coefficients in the series.

## 5. Why is finding a Fourier sine series important?

Fourier sine series are important in mathematics and science because they allow us to represent complex periodic functions using simpler trigonometric functions. This can be useful in analyzing and understanding the behavior of these functions, as well as in solving differential equations and other mathematical problems.

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