Is Calculating Only the Fourier Sine Series Sufficient for f(x) = x^3 on [-1,1]?

[mbud]

Homework Statement

Determine a general Fourier series representation for f(x) = x^3 -1<x<1

Homework Equations

The Attempt at a Solution

May seem like a stupid Q, but would i have to calculate a0, an, bn or since i know that x^3 is an odd function, could jump straight into calculating the Fourier sine series for odd functions. Would that give me a general representation?

 

[vishal007win]

looking at it one can directly jump to Fourier sine series
if you calculate all the terms initially assuming the complete Fourier representation you will find that coefficients associated with the even terms will go to zero.
so its more like using a known result.

 

[CompuChip]

Yes, you can restrict yourself to sines.

If you want you can explicitly check it, with an argument along the lines of
$$\int_{-1}^1 \cos(x) x^3 dx = \int_{-1}^0 \cos(x) x^3 dx + \int_0^1 \cos(x) x^3 dx = \int_0^1 \cos(-x) (-x)^3 dx + \int_0^1 \cos(x) x^3 dx = 0$$
because cos(-x) (-x)³ = - cos(x) x³

 

[mbud]

thanks.