Question

Find the Fourier series expansion of the periodic function defined on its fundamental cell as $$f(\theta)=\left\\{\begin{array}{ll} -\frac{1}{2}(\pi+\theta) & \text { if }-\pi \leq \theta<0 \\ \frac{1}{2}(\pi-\theta) & \text { if } 0<\theta \leq \pi\end{array}\right.$$

Short answer

The Fourier series expansion for the given function is \[f(\theta) = \sum_{n=1}^{\infty}\frac{1}{n}(\cos(n\pi) - 1)\sin(n\theta)\]

Step-by-step solution

Determine the Fourier Series components

The Fourier series expansion of a function \(f(\theta)\) is defined as: \[f(\theta) = a_0 + \sum_{n=1}^{\infty}(a_n\cos(n\theta) + b_n\sin(n\theta) )\]. Where \(a_n\) and \(b_n\) are Fourier coefficients. \(a_0\) is calculated as \(\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\theta)d\theta\), \(a_n\) is computed as \(\frac{1}{\pi}\int_{-\pi}^{\pi}f(\theta)\cos(n\theta)d\theta\) and \(b_n\) is calculated as \(\frac{1}{\pi}\int_{-\pi}^{\pi}f(\theta)\sin(n\theta)d\theta\). The task now is to compute the coefficients \(a_0, a_n\) and \(b_n\).

Compute the coefficient \(a_0\)

From the given periodic function \(f(\theta)\), we see that it is symmetric around the origin, thus \(a_0\) equals zero, because the integral from \(-\pi\) to \(\pi\) for a symmetric function around the origin equals 0

Compute the coefficient \(a_n\)

Because the function is odd, the cosine terms of the series will be all 0. Hence \(a_n = 0\). This is essentially due to the orthogonality property of \(\cos(n\theta)\) and \(\sin(n\theta)\) over the interval \([-π,π]\)

Compute the coefficient \(b_n\)

As \(f(\theta)\) is odd, we compute \(b_n\) on the entire interval from \(-\pi\) to \(\pi\). So \(b_n = \frac{1}{\pi}\int_{-\pi}^{0}-\frac{1}{2}(\pi+\theta)\sin(n\theta)d\theta + \frac{1}{\pi}\int_{0}^{\pi}\frac{1}{2}(\pi-\theta)\sin(n\theta)d\theta\). After a few calculations, this can be simplified to: \(b_n = \frac{1}{n}(\cos(n\pi) - 1)\) for \(n\neq0\)

Final answer

The Fourier series expression for \(f(\theta)\) becomes \[f(\theta) = \sum_{n=1}^{\infty}\frac{1}{n}(\cos(n\pi) - 1)\sin(n\theta)\]

Key concepts

Fourier Coefficients

Understanding Fourier coefficients starts with considering these as the specific numerical constants that uniquely define how a periodic function should be represented as a sum of trigonometric functions. When dealing with Fourier series, a periodic function could be any repetitive waveform, like a square wave or a triangular wave. Each coefficient corresponds to a particular sine or cosine term in the series, representing different 'harmonics' or frequency components of the function.

To determine them, you need to perform integration operations that consider the shape of the function over one period. Calculated carefully, these coefficients allow us to recreate the original function precisely using that sum of sine and cosine terms.

Periodic Functions

A periodic function is one that repeats its values at regular intervals, known as the period. The concept is commonly visualized through the sine and cosine functions that repeat every 2π radians. The importance of such functions in the context of Fourier series lies in their predictability and the ability to decompose them into simpler building blocks, i.e., a series of sine and cosine functions.

Periodicity is not just a mathematical curiosity, but it's found in various real-world signals like sound waves, alternating current, and light waves. By applying Fourier analysis to these functions, we can extract information about the frequency components, which is critical in fields like signal processing, communication, and acoustics.

Trigonometric Series

The Fourier series is a specific type of trigonometric series, which is essentially a series constructed of sine and cosine functions. When we talk about expanding periodic functions into their trigonometric series, we're trying to find a set of sine and cosine functions that, when summed together, will equal the original function over a period.

The coefficients of these trigonometric functions, found using the Fourier series method, are critical because they determine the amplitude or weight of each sine and cosine component. This decomposition into a trigonometric series is not just for theoretical analysis but forms the foundation for practical implementations, such as in electronic signal processing, where it is used to analyze frequency content and to synthesize signals.

Orthogonality Property

The concept of orthogonality plays a central role in determining Fourier coefficients because it pertains to the relationship between sine and cosine functions over a period. Orthogonality conveys that, over a full period, the integral of the product of different sine or cosine functions equals zero. This property is critical for simplifying the calculation of Fourier coefficients because it helps isolate each coefficient when integrating over the function's period.

Mathematically, we say that two functions are orthogonal if their integral product over a specific interval is zero. In the context of the Fourier series, this allows us to consider the contribution of each frequency component separately. It is particularly interesting in our exercise where the function's symmetry and orthogonality led to the simplification of coefficients, and ultimately, the Fourier series expansion.