Question

(i) Compute the Fourier coefficients of \(f(t)=e^{i n t}\) for a fixed \(n \in \mathbb{Z}\). (ii) Compute the Fourier coefficients of the \(2 \pi\)-periodic square wave which has \(f(t)=-1\) for \(-\pi \leq t<0\) and \(f(t)=1\) for \(0 \leq t<\pi\).

Short answer

(i) Fourier coefficients are \( c_k = \delta_{k,n} \). (ii) \( c_k = 0 \) for even \( k \), \( c_k = \frac{2}{k\pi} \) for odd \( k \).

Step-by-step solution

Define the Fourier Coefficients

Fourier coefficients for a function \( f(t) \) with period \( 2\pi \) are given by \( c_k = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-ikt} \, dt \). We need to compute these coefficients for the given functions.

Compute the Fourier Coefficients for \( f(t) = e^{i n t} \)

Substitute \( f(t) = e^{i n t} \) into the formula for Fourier coefficients: \[ c_k = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{i n t} e^{-ikt} \, dt = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{i(n-k)t} \, dt \].

Evaluate the Integral

The integral \( \int_{-\pi}^{\pi} e^{i(n-k)t} \, dt = \left[ \frac{1}{i(n-k)} e^{i(n-k)t} \right]_{-\pi}^{\pi} \). If \( n eq k \), the integral evaluates to 0; if \( n = k \), it evaluates to \( 2\pi \).

Conclude Fourier Coefficients for \( f(t) = e^{i n t} \)

The result from Step 3 implies \( c_k = \frac{1}{2\pi} \cdot 0 = 0 \) for \( n eq k \), and \( c_k = 1 \) for \( n = k \). Thus, \( c_k = \delta_{k,n} \) where \( \delta \) is the Kronecker delta.

Define the Square Wave Function

For the second part, we consider \( f(t) = \begin{cases} -1 & \text{if } -\pi \leq t < 0 \ 1 & \text{if } 0 \leq t < \pi \end{cases} \).

Compute Fourier Coefficient Formula for Square Wave

For the square wave, \[ c_k = \frac{1}{2\pi} \left( \int_{-\pi}^{0} (-1) e^{-ikt} \, dt + \int_{0}^{\pi} (1) e^{-ikt} \, dt \right) \].

Evaluate Integrals for Square Wave

The first integral evaluates to \( \frac{-1}{2\pi} \left[ \frac{1}{-ik} e^{-ikt} \right]_{-\pi}^{0} = \frac{1}{ik\pi} (e^{ik\pi} - 1) \) and the second integral evaluates to \( \frac{1}{2\pi} \left[ \frac{1}{-ik} e^{-ikt} \right]_{0}^{\pi} = \frac{1}{ik\pi} (1 - e^{-ik\pi}) \).

Simplify Result for Coefficients

Combine the evaluated integrals: \( c_k = \frac{1}{ik\pi} \left( (e^{ik\pi} - 1) + (1 - e^{-ik\pi}) \right) = \frac{1}{ik\pi} (2i \sin(k\pi)) \). Simplifying gives \( c_k = \begin{cases} 0 & \text{if } k \text{ is even} \ \frac{2}{k\pi} & \text{if } k \text{ is odd} \end{cases} \).

Conclude Fourier Coefficients for Square Wave

The Fourier coefficients for the square wave function are: \( c_k = 0 \) for even \( k \), and \( c_k = \frac{2}{k\pi} \) for odd \( k \).

Key concepts

Fourier coefficients

Fourier coefficients are a crucial component in Fourier Analysis, allowing us to decompose periodic functions into a sum of sines and cosines. A Fourier series uses these coefficients to represent a function as an infinite sum of harmonics. Given a function with period \(2\pi\), the Fourier coefficients \(c_k\) are calculated using the formula:

• \(c_k = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-ikt} \, dt\)

In this expression, \(k\) is an integer representing the harmonic number, collaborating with \(\omega\), the angular frequency, to evaluate each partial wave's contribution. By determining the Fourier coefficients, we can generate an approximation of the function by summing all its harmonic components. It is a powerful tool, especially when analyzing signals in electrical engineering, acoustics, and other fields that deal with periodic phenomena.

Square wave

A square wave is one of the most common periodic functions used in Fourier analysis. It alternates between two levels with a distinct sharp transition, creating a waveform that resembles a series of squares. Let's define a \(2\pi\)-periodic square wave function as:

• \(f(t) = -1\) for \(-\pi \leq t < 0\)
• \(f(t) = 1\) for \(0 \leq t < \pi\)

This characteristic pattern presents a challenge for Fourier analysis as it contains only odd harmonics. To find the Fourier coefficients for a square wave, the function is represented and integrated over one period. The results reveal that:

• \(c_k = 0\) for even \(k\)
• \(c_k = \frac{2}{k\pi}\) for odd \(k\)

This coefficient pattern demonstrates that a square wave is just the sum of all odd harmonic components, a feature that significantly contrasts from smooth waves like sine and cosine.

Kronecker delta

In mathematics, the Kronecker delta function \(\delta_{i,j}\) is a helpful concept when dealing with discrete variables and sums. It is a special case function that evaluates to:

• 1 when \(i = j\)
• 0 when \(i eq j\)

Within the context of Fourier coefficients, the Kronecker delta function simplifies expressions by denoting when two indices are equal, filtering unwanted harmonics, and highlighting specific frequencies. For example, in Fourier Analysis, this principle proves useful in identifying the specific frequency components that remain in a Fourier series, making it easier to handle functions represented by discrete data points.

Fourier series

The Fourier series is foundational to Fourier analysis. It allows us to express a complex periodic function as an infinite sum of sinusoidal functions (sines and cosines). This decomposition is crucial in analyzing signal processing, acoustics, and even heat distribution problems. The general form of a Fourier series is:

• \(f(t) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(nt) + b_n \sin(nt)]\)

Here, \(a_0\), \(a_n\), and \(b_n\) are determined using Fourier coefficients, which capture the magnitude and phase for each harmonic frequency. The Fourier series transforms a periodic function into a combination of elementary trigonometric functions, enabling deeper insights into its frequency content and characteristics. Transforming a challenging function into an infinite sum of simple waves is a powerful technique facilitating a wide array of technological advancements.

Periodic functions

Periodic functions repeat their values in regular intervals, characterized by a constant period \(T\). Typical examples include sine waves, square waves, and triangular waves. For a function \(f(t)\), periodicity means that:

• \(f(t+T) = f(t)\) for all \(t\)

Periodic functions are critical for Fourier analysis because they can be expressed as a Fourier series. This characteristic makes them highly applicable in modeling repetitive phenomena such as vibrations, sound waves, or signals. The periodic nature assures that their analysis involves recurring patterns that can be broken down and understood through their harmonic components, proving invaluable in engineering, physics, and applied mathematics.