Fourier Series of f(x) x - x^2 - x in [0, 2]: A Comprehensive Guide

Understanding the Fourier series of a function is crucial in various fields of mathematics and engineering, especially when dealing with periodic functions. In this detailed guide, we will explore the Fourier series representation of the function f(x) x - x^2 - x over the interval [0, 2]. We will use the Fourier series definition and compute the necessary coefficients to represent this function.

1. Introduction to the Fourier Series

The Fourier series is a way to represent a periodic function as a sum of sine and cosine waves. For a function f(x) defined over the interval [0, L], the Fourier series is given by:

f(x) ≈ frac{a_0}{2} sum_{n1}^{∞} [a_n cos(frac{nπx}{L}) b_n sin(frac{nπx}{L})]

2. Compute the Fourier Coefficients for f(x) x - x^2 - x in [0, 2]

Given the function f(x) x - x^2 - x, we will determine the Fourier coefficients a_n and b_n.

2.1 Compute a_0

The constant term a_0 is calculated using the following integral:

a_0 frac{1}{L} int_0^L f(x) dx

In this case, L 2, so we have:

[a_0 frac{1}{2} int_0^2 (x - x^2 - x) dx int_0^2 (2x - 3x^2) dx]

First, compute the integral: [a_0 left[x^2 - x^3right]_0^2 4 - 8 -4]

Alternatively, using the interval-inverting substitution by replacing x with 2 - x we have:

[a_0 - int_0^2 (2 - xx - 1x) dx -a_0]

This implies that a_0 0.

2.2 Compute a_n

To compute the coefficients a_n, we use the following integral:

[a_n frac{1}{L} int_0^L f(x) cos(frac{nπx}{L}) dx]

For our interval L 2, this gives:

[a_n int_0^2 (2x - 3x^2) cos(frac{nπx}{2}) dx]

Using the interval-inverting substitution again: [a_n - int_0^2 (2 - xx - 1x) cos(frac{nπ2 - x}{2}) dx -a_n] This implies that a_n -a_n, leading to a_n 0 for all n.

2.3 Compute b_n

For the sine coefficients b_n, we use the following integral:

[b_n frac{1}{L} int_0^L f(x) sin(frac{nπx}{L}) dx]

Again, for L 2, this becomes:

[b_n int_0^2 (2x - 3x^2) sin(frac{nπx}{2}) dx]

Using integration by parts, we get: [b_n left[2x - 3x^2right] left(-frac{cos(nπx/2)}{nπ}right) - 2x left(-frac{6sin(nπx/2)}{nπ^2}right) - 3x^2 left(-frac{6cos(nπx/2)}{nπ^3}right)] Evaluating this from 0 to 2, we obtain: [b_n frac{12}{nπ^3}]

Hence, the Fourier series for f(x) x - x^2 - x in [0, 2] can be expressed as:

[f(x) sim sum_{n1}^{∞} frac{12}{nπ^3} sin(frac{nπx}{2})]

3. Conclusion

The given function f(x) x - x^2 - x can be represented as a Fourier sine series, highlighting the importance of integrating by parts and utilizing symmetry in the computation of the coefficients. Understanding these coefficients and their computation provides a deeper insight into the nature of periodic functions and their series representation.

4. Related Keywords

Fourier series Function expansion Integration by parts

5. References

[1] Euler, L. Istitutiones calculi differentialis. Petropoli, 1775.

[2] Fourier, J. The Analytical Theory of Heat. Cambridge University Press, 1878.