Question

In Problems \(5.1\) to \(5.8\), define each function by the formulas given but on the interval \((-l, l)\). [That is, replace \(\pm \pi\) by \(\pm l\) and \(\pm \pi / 2\) by \(\pm 1 / 2\).) Expand each function in a sine-cosine Fourier series and in a complex exponential Fourier series.

Short answer

Replace limits, find coefficients using integrals, write series.

Step-by-step solution

- Understand the Interval

The exercise defines the interval as \((-l, l)\). This means we will replace the limits \( \pm \pi \) by \( \pm l \) and \( \pm \pi/2 \) by \( \pm 1/2 \). This establishes our domain for the Fourier series.

- Identify Fourier Series

A Fourier series expansion of a function can be represented in two forms: sine-cosine Fourier series and complex exponential Fourier series. The general formulas for these series are given by: \[f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos \frac{n \pi x}{l} + b_n \sin \frac{n \pi x}{l}) \] for sine-cosine series and \[f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i \frac{n \pi x}{l}} \] for complex exponential series.

- Calculate Coefficients for Sine-Cosine Series

To find the coefficients \(a_0\), \(a_n\), and \(b_n\): \[a_0 = \frac{1}{l} \int_{-l}^{l} f(x) \, dx \] \[a_n = \frac{1}{l} \int_{-l}^{l} f(x) \cos \frac{n \pi x}{l} \, dx \] \[b_n = \frac{1}{l} \int_{-l}^{l} f(x) \sin \frac{n \pi x}{l} \, dx \] Perform these calculations for each function as defined in Problems 5.1 to 5.8.

- Calculate Coefficients for Complex Exponential Series

To find the coefficients \(c_n\): \[c_n = \frac{1}{2l} \int_{-l}^{l} f(x) e^{-i \frac{n \pi x}{l}} \, dx \] Perform this calculation for each function as defined in Problems 5.1 to 5.8.

- Write the Final Fourier Series

Using the computed coefficients for each function from Problems 5.1 to 5.8, write the final sine-cosine Fourier series and the complex exponential Fourier series in their respective forms.

Key concepts

Sine-Cosine Fourier Series

The sine-cosine Fourier series is a way of representing a function as a sum of sines and cosines. This method is particularly useful when dealing with periodic functions. The general form of the sine-cosine Fourier series is: