Introduction to Fourier Sine Series

Before diving into the detailed calculation of the Fourier sine series for the function f(x) x^2 defined on the interval [0, L], it is essential to understand the definition of a Fourier Sine Series.

Definition of Fourier Sine Series

A Fourier sine series represents a function f(x) defined on the interval [0, L] as a sum of sine functions with coefficients determined by the integral of the function multiplied by sine functions over the interval [0, L]. The series takes the form:

f(x) x^2 can be represented as

where the coefficients b_n are given by

Derivation of Fourier Sine Series Coefficients for f(x) x^2

To determine the Fourier sine series for f(x) x^2 on [0, L], we need to calculate the coefficients b_n

The coefficients b_n are calculated as:

Breaking this integral into parts, we get:

Next, we calculate each of these integrals separately.

First Integral:

Using integration by parts, let:

u x, dv sin left(frac{n pi x}{L}right) dx and hence du dx, v -frac{L}{n pi} cos left(frac{n pi x}{L}right)

Then:

The second integral is:

Since sin(0) 0 and sin(pi n) 0 for any integer n, the second integral evaluates to zero.

Thus:

Second Integral:

This integral can be evaluated directly:

Again, since cos(pi n) -1 and cos(0) 1, we get:

Substituting back, we find:

After further simplification:

Final Fourier Sine Series

Putting it all together:

Conclusion

This series represents the function x^2 defined on the interval [0, L] as a sum of sine functions with coefficients that have been derived through the integral calculations.

Additional Resources

For more detailed information, you can refer to the following resources:

Wikipedia: Fourier Series Online tutorials on Fourier Series and Fourier Sine Series