Question

A Berechnen Sie die Fourierreihe der Funktion \(f(t)=f(t+4 \pi)\) mit $$ f(t)=\left\\{\begin{array}{lll} 0 & \text { für } & -2 \pi \leq t<-\pi \\ 1 & \text { für } & -\pi \leq t<\pi \\ 0 & \text { für } & \pi \leq t<2 \pi \end{array}\right. $$ B Berechnen Sie die Fourierreihe für die Funktion $$ f(t)=\left\\{\begin{array}{lll} -1 & \text { für } & -\pi \leq t<0 \\ +1 & \text { für } & 0 \leq t \leq \pi \end{array}\right. $$

Short answer

The Fourier series are: \(f(t) = \frac{4}{\pi} \sum_{n=1,3,5,...}^\infty \frac{ \sin( n t/2) }{n}\) for the first function and \(f(t) = \frac{4}{\pi} \sum_{n=1,3,5,...}^\infty \frac{ \sin( n t) }{n}\) for the second function.

Step-by-step solution

Identify Period and Fourier Coefficients for First Function

The function \(f(t)=f(t+4 \pi)\) is periodic with period \(T=4\pi\). Its Fourier series has the form \(f(t) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(n \omega t) + b_n \sin(n \omega t)]\), with \(\omega = 2\pi/T = 1/2\) and coefficients calculated as: \(a_0 = \frac{1}{T} \int_{-T/2}^{T/2} f(t) dt\), \(a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(t) \cos(n \omega t) dt\), \(b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(t) \sin(n \omega t) dt\).

Calculate Fourier Coefficients for First Function

Applying the integral over the intervals where \(f(t) \neq 0\) we find: \(a_0 = 0\), \(a_n = 0\) and \(b_n = \frac{1}{n\pi} [1-(-1)^n]\) for \(n\geq 1\).

Write Fourier Series for First Function

We substitute the coefficients into the Fourier series to obtain: \(f(t) = \frac{4}{\pi} \sum_{n=1,3,5,...}^\infty \frac{ \sin(n t/2) }{n}\). The series converges to one on \(-\pi \leq t<\pi\) and to zero otherwise.

Identify Period and Fourier Coefficients for Second Function

The function \(f(t)=\left\{\begin{array}{lll} -1 & \text { for } & -\pi \leq t<0 \ +1 & \text { for } & 0 \leq t \leq \pi \end{array}\right.\) is periodic with period \(T=2\pi\). The Fourier series and the coefficients are calculated in the same way as in step 1.

Calculate Fourier Coefficients for Second Function

We find: \(a_0 = 0\), \(a_n = 0\) and \(b_n = \frac{2}{n} [1-(-1)^n]\) for \(n\geq 1\).

Write Fourier Series for Second Function

We obtain: \(f(t) = \frac{4}{\pi} \sum_{n=1,3,5,...}^\infty \frac{ \sin(n t) }{n}\). This series converges to one on \(0 \leq t<\pi\) and to minus one on \(-\pi \leq t<0\).

Key concepts

Periodic Functions

A periodic function is a function that repeats its values in regular intervals or periods. For instance, the function given in the exercise, \(f(t) = f(t+4\pi)\), is periodic with a period \(T = 4\pi\). This means that every \(4\pi\) units along the \(t\)-axis, the function outputs repeat their pattern.
This concept is essential because it helps simplify complex functions by reducing them to a basic repeating unit.

• Period: The smallest interval after which the function begins to repeat.
• Cyclic Nature: Allows complex waveforms to be analyzed and understood through simpler, repeating intervals.

Understanding the nature of periodic functions is crucial in Fourier analysis, as the Fourier series aims to represent these functions as sums of simpler sine and cosine functions.

Fourier Coefficients

Fourier coefficients are key components used to build the Fourier series of a function. These coefficients determine the specific weights of the sine and cosine terms that combine to approximate a periodic function. The coefficients are calculated using integrals over one period of the function.
For a function \(f(t)\) with period \(T\), the Fourier series is represented as: \[f(t) = a_0 + \sum_ {n=1}^{\infty} \left[a_n \cos(n \omega t) + b_n \sin(n \omega t)\right],\]where:

• \(a_0\) is the average value of the function over one period, calculated as \(a_0 = \frac{1}{T} \int_{-T/2}^{T/2} f(t) \, dt\).
• \(a_n\) quantifies the contribution of the cosine terms, found by evaluating \(a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(t) \cos(n \omega t) \, dt\).
• \(b_n\) represents the weights of the sine terms, calculated from \(b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(t) \sin(n \omega t) \, dt\).

These coefficients are fundamental to decomposing periodic functions into their frequency components, making them a powerful tool in mathematical physics and signal processing.

Trigonometric Series

The trigonometric series is the sum of sine and cosine terms used to represent a function. This series is crucial in forming the Fourier series, which expresses periodic functions in terms of trigonometric functions. For example, the challenge of the exercise is to express given functions as trigonometric series, where each term corresponds to a specific frequency component of the original function.
Consider the series derived from the first function in the problem:\[f(t) = \frac{4}{\pi} \sum_{n=1,3,5,...}^{\infty} \frac{ \sin(n t/2) }{n}\]The terms in this series account for different harmonics—or integer multiples of a fundamental frequency—reflecting the function's periodic nature.

• The terms with sine capture the function's odd symmetry around the origin.
• The presence of only sine terms, and their non-existence at even harmonics, is due to the odd nature of the function.

Each component of the trigonometric series contributes to faithfully reconstructing the entire function over its period.

Mathematical Physics

In mathematical physics, concepts like the Fourier series and trigonometric functions are indispensable tools for solving a wide range of problems. These mathematical strategies convert complex wave-like structures into manageable sinusoidal forms.
This conversion is immensely helpful in fields such as:

• Signal Processing: Where signals can be broken down into frequency components for analysis or compression.
• Heat Transfer: In problems where solutions like the heat equation reveal longitudinal temperature distributions over time.
• Wave Equation Solutions: Simplifying the representation of electromagnetic or sound waves.

In physics, using Fourier series can significantly simplify solving differential equations, explaining oscillations, and studying resonance phenomena. It's a bridge between mathematical theory and physical application, making complex phenomena easier to solve and understand.