Question

(a) Bestimmen Sie das komplexe FOURIER-Polynom \(n\)-ter Ordnung der Funktion \(f\), die definiert ist durch \(f(x)=e^{2 x}\) für \(0 \leq x \leq 1\) mit \(f(x)=f(x+2)\) und \(f(x)=f(-x)\). (b) Wie lautet die zugehörige Darstellung des FOURIER-Polynoms \(n\)-ter Ordnung im Reellen?.

Short answer

The complex Fourier polynomial is expressed with coefficients \( c_k = \frac{e^{2 - i\pi k} - 1}{2 - i\pi k} \). The real form involves sines and cosines.

Step-by-step solution

Understanding the Function Periodicity

The function given is periodic with period 2, meaning that it repeats every 2 units. This allows us to express it using a Fourier series, which accommodates its periodic nature.

Expand the Function in a Complex Fourier Series

For a function periodic with period 2, the complex Fourier series is given by:\[ f(x) = \sum_{k=-\infty}^{\infty} c_k e^{i\frac{\pi k x}{1}} \]where coefficients \(c_k\) are computed as:\[ c_k = \frac{1}{T} \int_0^T f(x) e^{-i\frac{\pi k x}{1}} \, dx \]For our periodic function, we evaluate:\[ c_k = \int_0^1 e^{2x} e^{-i\pi k x} \, dx \]

Calculate the Fourier Coefficients

Evaluate the integral for \(c_k\):\[ c_k = \int_0^1 e^{(2 - i\pi k)x} \, dx = \left[ \frac{1}{2 - i\pi k} e^{(2 - i\pi k)x} \right]_0^1 \]\[ c_k = \frac{e^{2 - i\pi k} - 1}{2 - i\pi k} \]

Construct the Complex Fourier Polynomial

Use the coefficients to construct the truncated complex Fourier series:\[ P_n(x) = \sum_{k=-n}^{n} \frac{e^{2 - i\pi k} - 1}{2 - i\pi k} e^{i\pi k x} \]

Convert the Complex Fourier Series to Real Form

To obtain the Fourier series in real form, separate the complex exponential into sine and cosine components.Using Euler's formula, \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \), rewrite the terms and simplify accordingly to get an expression involving only real parts, using symmetry properties where applicable.

Key concepts

Complex Fourier Series

A complex Fourier series is a way to represent periodic functions using complex exponentials. When you have a periodic function, like the one in the problem with a period of 2, it can be represented by a sum of complex exponential terms. This is particularly useful because complex exponentials can easily express oscillations, which are inherent in periodic functions.

For a function with period 2, the complex Fourier series takes the form:

• \( f(x) = \sum_{k=-\infty}^{\infty} c_k e^{i\frac{\pi k x}{1}} \)

The coefficients \( c_k \) are what "weight" each of these oscillations to match the function you're analyzing. Calculating these coefficients involves integrating the product of your function and the complex exponential over one period of the function. In our case, this translates to an interval from 0 to 1.

Periodicity of Functions

Periodicity refers to the property of a function repeating its values at regular intervals. In the exercise, our function is noted to be periodic with a period of 2. This means that for any value of \( x \), the function satisfies \( f(x) = f(x+2) \).

Understanding the periodicity is crucial because it defines how we set up the Fourier series. The periodicity helps identify that the function can be expressed by a sum of sines and cosines (or in complex form, sine and cosine components through Euler's formula). It ensures that the function's behavior over one interval (in this case, 0 to 1) is replicated over all other intervals, allowing us to use integrals that span just one complete period when computing Fourier coefficients.

Fourier Coefficients

Fourier coefficients are the essential ingredients of a Fourier series. They dictate the amplitude and phase of each sine or cosine wave (or complex exponential) in the series. For the exercise, we compute the coefficients \( c_k \) using the formula:

• \( c_k = \int_0^1 e^{(2 - i\pi k)x} \ dx = \left[ \frac{1}{2 - i\pi k} e^{(2 - i\pi k)x} \right]_0^1 \)

These coefficients are calculated by integrating over one period of the function's interval, here from 0 to 1. This means they contain all necessary information about how the function behaves over its periodic segment. The knowledge of these coefficients allows you to construct the function's approximation using a finite number of terms (a Fourier polynomial). Whenever you compute these coefficients, you essentially "decode" how a simple periodic function is built through its angular constants.

Real and Imaginary Parts in Fourier Series

In a complex Fourier series, each term consists of a complex exponential, which can be broken down into real and imaginary parts. This is done using Euler's formula: \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \).

The exercise also requires us to express the computed complex Fourier series in real terms. To achieve this, each \( e^{i\pi k x} \) in the series can be rewritten as:

• Real Part: \( \cos(\pi k x) \)
• Imaginary Part: \( i\sin(\pi k x) \)

This transformation is useful when you need to examine the series without involving complex numbers. The real part typically relates to the cosine terms, and the imaginary part to the sine terms of the Fourier series. By doing this, the Fourier polynomial can be represented entirely using real functions, making it more applicable in contexts where real number computations are necessary, such as engineering applications or practical physics problems.