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Chapter 2: Problem 3

Gegeben sei dic \(2 \pi\)-periodische Funktion \(f\) durch $$ f(x)= \begin{cases}\frac{1}{\pi^{3}} x^{4}-\frac{3}{2 \pi} x^{2}+\frac{5 \pi}{16} & \text { f眉r }|x|<\frac{\pi}{2} \\ \cos x & \text { f眉r } \frac{\pi}{2} \leq|x| \leq \pi\end{cases} $$ a) Skizzieren Sie den Graphen von \(f\). b) Beweisen Sie, da脽 \(f\) auf \(\mathbb{R}\) genau zweimal stetig differenzicrbar ist. c) Berechnen Sie die Fourier-Reihe von \(f\).

Short Answer

Expert verified
(a) Sketch a piecewise function with both polynomial and cosine parts. (b) \( f \) is twice differentiable at \( x = \pm \frac{\pi}{2} \). (c) Calculate Fourier coefficients from integrals and construct series.

Step by step solution

01

Understanding the function

The function \( f(x) \) is defined in two parts: a polynomial function for \( |x| < \frac{\pi}{2} \), and a cosine function for \( \frac{\pi}{2} \leq |x| \leq \pi \). The function is \( 2\pi \)-periodic, meaning it repeats its values every \( 2\pi \).
02

Sketching the graph

For \( |x| < \frac{\pi}{2} \), sketch the graph of \( f(x) = \frac{1}{\pi^{3}} x^{4} - \frac{3}{2 \pi} x^{2} + \frac{5\pi}{16} \), which is a quartic polynomial. For \( \frac{\pi}{2} \leq |x| \leq \pi \), sketch the graph of \( f(x) = \cos x \). Since it is \( 2\pi \)-periodic, repeat these sketches every \( 2\pi \).
03

Identifying points of differentiability

To find where \( f(x) \) is twice continuously differentiable, we examine the points \( x = \pm \frac{\pi}{2}, \pm \pi \) since these are where the piecewise definitions meet. Compute left- and right-hand derivatives up to the second order at these critical points to verify differentiability.
04

Check differentiability at \( x = \pm\frac{\pi}{2} \)

Calculate the first and second derivatives of \( f(x) \) near \( x = \pm\frac{\pi}{2} \) from both sides. Show that these derivatives match for continuity and twice differentiability.
05

Check differentiability at \( x = \pm\pi \)

Compute the first and second derivatives where \( f(x) \) meets its period (\( x = \pm\pi \)). Check continuity of separate derivatives again to ensure \( f(x) \) remains twice differentiable.
06

Writing the Fourier series

To calculate the Fourier series of \( f(x) \), compute the coefficients using the formula: \[ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \, dx \]\[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \]\[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \]Calculate these integrals by splitting the interval from \(-\pi\) to \(\pi\) where \( f(x) \) changes definition.
07

Evaluate the integrals and build the series

Solve the integrals for \( a_0, a_n, \) and \( b_n \) to obtain numerical values. Assemble the Fourier series \( f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) \) using these coefficients.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Periodic Function
A periodic function is one that repeats its values at regular intervals or periods. In the given exercise, the function \( f(x) \) is defined as a \( 2\pi \)-periodic function. This means that every \( 2\pi \) units along the x-axis, the function will start to repeat its pattern. Understanding periodicity is crucial when working with Fourier series, as it allows the function to be expressed as a sum of sine and cosine terms which are inherently periodic.
  • The period here is \( 2\pi \), common in trigonometric functions like sine and cosine.
  • Periodic functions are essential in representing waveforms and oscillations in physics and engineering.
  • The periodic nature facilitates the prediction of the function's values outside its initial segment.
Recognizing a function's period helps simplify analysis and calculations, especially when dealing with integrals over one full period.
Piecewise Functions
Piecewise functions are defined by different expressions over different intervals. For this exercise, the function \( f(x) \) is defined by a polynomial expression for \( |x| < \frac{\pi}{2} \) and a trigonometric expression for \( \frac{\pi}{2} \leq |x| \leq \pi \). Such functions are typical in modeling scenarios where a single rule doesn't apply across the entire domain.
  • They address changes in function behavior across different intervals.
  • Piecewise functions allow complex models that can handle abrupt changes.
  • They are key in solving real-world problems like modeling transactions or service rates.
When handling piecewise functions, each part needs to be examined individually, especially at the boundaries, where the definitions switch. This is crucial when checking for continuity and differentiability.
Differentiability
Differentiability is the property that allows us to find a derivative for a function at a point. For \( f(x) \) in this exercise, we need to check differentiability at critical points where the function switches definitions, specifically at \( x = \pm \frac{\pi}{2} \) and \( x = \pm \pi \). This involves ensuring that the left and right derivatives match at these points.
  • Differentiability at a point ensures the function is smooth and can be approximated by a tangent line.
  • Checking for differentiability involves calculating the first and second derivatives.
  • Ensuring the derivatives are continuous provides evidence for twice differentiability.
Identifying whether a function is differentiable is necessary for determining the smoothness and behavior of the function, which is especially important for calculating Fourier series accurately.
Polynomial and Trigonometric Functions
In this exercise, the function \( f(x) \) features both polynomial and trigonometric components. These are two of the most common types of functions encountered in calculus and Fourier analysis.
  • Polynomials are functions composed of terms with variables raised to integer powers. Here, the polynomial part is a quartic function for \( |x| < \frac{\pi}{2} \).
  • Trigonometric functions, like \( \cos x \), are based on angles and can represent periodic behavior. For this function, \( \cos x \) handles the interval \( \frac{\pi}{2} \leq |x| \leq \pi \).
  • Combining these functions provides versatility and flexibility in modeling.
Understanding how to manipulate and integrate these functions is foundational in building and evaluating Fourier series, which is a powerful tool in both pure and applied mathematics.

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Most popular questions from this chapter

Mit Hilfe des modifizierten Horner-Schemas sind die ersten acht Glieder (bis \(x^{7}\) ) der Potenzreihe mit Entwicklungspunkt 0 der folgenden Funktionen \(z u\) berechnen. a) \(f(x)=\frac{\cos x}{\cosh x}\) b) \(f(x)=e^{-x} \cos x\) c) \(f(x)=\frac{x}{\sin x}\); d) \(f(x)=x^{2} \cot x\)

Durch Potenzreihenentwicklung sind folgende Grenzwerte zu berechnen: a) \(\lim _{x \rightarrow \infty} x \cdot \ln \frac{x+3}{x-3}\); b) \(\lim _{x \rightarrow 0} \frac{x-\sin x}{x^{2} \sin x}\) c) \(\lim _{x \rightarrow 0} \frac{\mathrm{e}^{x^{2}-x}+x-1}{1-\sqrt{1-x^{2}}}\); d) \(\lim _{x \rightarrow 0} \frac{2 \sqrt{1+x^{2}}-x^{2}-2}{\left(\mathrm{c}^{x^{2}}-\cos x\right) \sin \left(x^{2}\right)}\)

Mit Hilfe einer Partialbruchzerlegung gebe man die Potenzreihenentwicklung mit einem geeigneten Entwicklungspunkt von folgenden Funktionen an: a) \(f(x)=\frac{x}{x^{2}+x-2}\) b) \(f(x)=\frac{1}{x^{2}+4 x+5}\) c) \(f(x)=\frac{5-2 x}{6-5 x+x^{2}}\) d) \(f(x)=\frac{11-9 x}{6+x-12 x^{2}}\)

a) Wie lautet die Fourier-Reihe der Funktion \(f\) mit \(f(x)=|\cos x| ?\) b) Beweisen Sie mit Hilfe von a) die Beziehung $$ \frac{1}{1 \cdot 3}-\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}-+\cdots=\frac{\pi}{4}-\frac{1}{2} $$

Entwickeln Sie die folgenden Funktionen in eine Potenzreihe mit dem Entwicklungspunkt Null, und geben Sie den Konvergenzradius an. a) \(f(x)=\frac{1}{(1+x)^{2}}\) b) \(f(x)=\frac{\ln (1+x)}{1+x}\) c) \(f(x)=e^{\sin x}\) d) \(f(x)=\cos ^{2} x\) e) \(f(x)=\mathrm{e}^{x} \cdot \sin x\) f) \(f(x)=\sqrt{\frac{1+x}{1-x}}\)
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