Question

Find the Fourier series of each of the following functions. (a) \(f(x)=|\sin x|\) (b) \(f(x)= \begin{cases}0, & -\pi \leq x \leq 0 \\ e^{x}, & 0

Short answer

The Fourier series for \(|\sin x|\) is \(\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{(-1)^k}{2k+1}\cos((2k+1)x)\).

Step-by-step solution

Analyze the function

The function can be observed for its periodic properties. Since the function repeats every period of length \(2\pi\), it can be expanded using Fourier series.

Determine coefficients for cosine terms

We need to calculate the Fourier cosine coefficients, \(a_0\) and \(a_n\), for the function \(f(x) = |\sin x|\). Use the formula: \(a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} |\sin x| \, dx\) and \(a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} |\sin x| \cos(nx) \, dx\).

Calculate sine coefficient

For odd functions, typically only sine coefficients, \(b_n\), are non-zero. We calculate it using \(b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} |\sin x| \sin(nx) \, dx\), although in this specific even function is not needed.

Evaluate integrals for coefficients

Calculate the integrals for each coefficient. The integration will use properties of sine and cosine functions, often resulting in simplified forms due to periodic properties.

Formulate Fourier series

Substitute these coefficients back into the Fourier series formula: \( f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} [ a_n \cos(nx) + b_n \sin(nx) ] \). For \(|\sin x|\) the final Fourier series becomes something like \( \frac{4}{\pi} \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \cos((2k+1)x) \).

Key concepts

Fourier coefficients

Fourier coefficients are the backbone of Fourier series. They allow us to break down complex periodic functions into sums of simpler trigonometric functions. There are two main types of Fourier coefficients: the cosine coefficients, denoted as \(a_n\), and the sine coefficients, denoted as \(b_n\). To compute these coefficients for a function \(f(x)\), we use integrals over a specific interval, typically \([-\pi, \pi]\). For example:

• Cosine coefficient \(a_0\) is given by \(a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx\).
• Similarly, \(a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx\) for \(n \geq 1\).
• Sine coefficient \(b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx\).

These coefficients give us insight into the function's periodic behavior and its symmetry properties. Typically, if the function is even, only the cosine terms contribute to the Fourier series.

cosine terms

Cosine terms in a Fourier series are significant for capturing the even components of a periodic function. They are expressed as \(a_n \cos(nx)\), where \(n\) is a positive integer. The term \(a_0/2\) often appears at the beginning of the series and represents the average or mean value of the function over one period.Determining cosine coefficients, \(a_n\), involves evaluating the integral \(a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx\). This highlights the function's symmetry around the y-axis:

• If \(f(x)\) is even, its Fourier series will mostly feature cosine terms.
• This is because the product of two even functions (like two cosine terms) is even.
• Hence, only cosine terms are applicable for functions where \(f(x) = f(-x)\).

Cosine terms help form the backbone of a Fourier series, especially for even functions like \(|\sin x|\).

sine terms

Sine terms in a Fourier series relate to the odd parts of a periodic function. They appear as \(b_n \sin(nx)\), where \(n\) is a positive integer. Calculating sine coefficients, \(b_n\), involves integrating the function multiplied by the sine term over its period:\[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \] Sine terms are particularly prevalent in the series of odd functions:

• Odd functions satisfy \(f(x) = -f(-x)\).
• The product of an even and an odd function, or two odd functions, results in an odd function.
• Thus, in a Fourier series of an odd function, only sine terms are significant.

For example, a function like \(f(x) = \sin x\) would primarily use sine terms in its Fourier series. However, for an even function like \(|\sin x|\), sine terms generally do not appear.

trigonometric functions

Trigonometric functions, namely sine and cosine, form the foundation of Fourier series. These functions are periodic, meaning they repeat their values over set intervals, which aligns perfectly with the periodic nature of Fourier series.

• **Sine Functions:** Represented by \( \sin(x) \), these functions have a wave-like structure that starts at zero and completes a full cycle from \(0\) to \(2\pi\).
• **Cosine Functions:** Represented by \( \cos(x) \), these also have a wave-like pattern but start at a maximum point.
• Combining these trigonometric functions, Fourier series can accurately describe even the most complex periodic signals.

The interplay between sine and cosine within Fourier series enables the transformation of complex expressions into manageable and iterable forms. This plays a crucial role in many fields such as engineering, physics, and signal processing.