To determine the components of the Fourier series expansion of a function, we first need to understand the concept of even and odd functions and how they relate to Fourier series coefficients.

Fourier Series Expansion of x³

The Fourier series allows us to represent a periodic function as a sum of sines and cosines. For a function \(f(x)\) defined on the interval \([-L, L]\) with periodic continuation, the Fourier series is given by:

\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right) \]

where the coefficients are calculated as:

• \( a_0 = \frac{1}{L} \int_{-L}^{L} f(x) dx \)
• \( a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx \)
• \( b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx \)

Understanding Even and Odd Functions

The nature of the function (whether it's even or odd) significantly simplifies the calculation of these coefficients:

• A function \(f(x)\) is considered even if \(f(-x) = f(x)\). Examples include \(x^2\), \(\cos(x)\). For an even function on a symmetric interval \([-L, L]\), \(a_0\) and \(a_n\) coefficients may be non-zero, while all \(b_n\) coefficients (sine terms) are zero.
• A function \(f(x)\) is considered odd if \(f(-x) = -f(x)\). Examples include \(x^3\), \(\sin(x)\). For an odd function on a symmetric interval \([-L, L]\), \(a_0\) and \(a_n\) coefficients (constant and cosine terms) are zero, while \(b_n\) coefficients (sine terms) may be non-zero.
• If a function is neither even nor odd, its Fourier series will generally contain both sine and cosine terms, and potentially a constant term.

Analyzing the Function f(x) = x³

We are given the function \(f(x) = x^3\) in the interval \(-1 \le x < 1\). This interval is symmetric around zero, with \(L=1\). Let's check the parity of the function \(f(x) = x^3\):

Substitute \(-x\) into the function:

\[ f(-x) = (-x)^3 = -x^3 \]

Compare \(f(-x)\) with \(f(x)\):

We observe that \(f(-x) = -f(x)\). This condition defines an odd function.

Fourier Series Coefficients for Odd Functions

Since \(f(x) = x^3\) is an odd function over the symmetric interval \([-1, 1]\), we can determine which coefficients will be zero without explicit calculation:

• The constant term \(a_0\) is zero because the integral of an odd function over a symmetric interval is always zero: \[ a_0 = \frac{1}{L} \int_{-L}^{L} f(x) dx = 0 \]
• The cosine coefficients \(a_n\) are zero because the product of an odd function \(f(x)\) and an even function \(\cos\left(\frac{n\pi x}{L}\right)\) results in an odd function. The integral of an odd function over a symmetric interval is zero: \[ a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx = 0 \]
• The sine coefficients \(b_n\) are generally non-zero because the product of an odd function \(f(x)\) and an odd function \(\sin\left(\frac{n\pi x}{L}\right)\) results in an even function. The integral of an even function over a symmetric interval is generally non-zero: \[ b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx \] For \(f(x) = x^3\) and \(L=1\), the \(b_n\) coefficients would be \(b_n = 2 \int_{0}^{1} x^3 \sin(n\pi x) dx\), which would yield non-zero values.

Conclusion

Since \(f(x) = x^3\) is an odd function over the interval \(-1 \le x < 1\), its Fourier series expansion will only contain sine terms. The constant term and all cosine terms will be zero.