Problem 16 - Advanced Engineering Mathematics

Question

$$\begin{array}{l} a_{0}=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) d x=\frac{1}{\pi} \int_{0}^{\pi}\left(e^{x}-1\right) d x=\frac{1}{\pi}\left(e^{\pi}-\pi-1\right) \\\ a_{n}=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos n x d x=\frac{1}{\pi} \int_{0}^{\pi}\left(e^{x}-1\right) \cos n x d x=\frac{\left[e^{\pi}(-1)^{n}-1\right]}{\pi\left(1+n^{2}\right)} \\ b_{n}=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin n x d x=\frac{1}{\pi} \int_{0}^{\pi}\left(e^{x}-1\right) \sin n x d x=\frac{1}{\pi}\left(\frac{n e^{\pi}(-1)^{n+1}}{1+n^{2}}+\frac{n}{1+n^{2}}+\frac{(-1)^{n}}{n}-\frac{1}{n}\right) \\\ f(x)=\frac{e^{\pi}-\pi-1}{2 \pi}+\sum_{n=1}^{\infty}\left[\frac{e^{\pi}(-1)^{n}-1}{\pi\left(1+n^{2}\right)} \cos n x+\left(\frac{n}{1+n^{2}}\left[e^{\pi}(-1)^{n+1}+1\right]+\frac{(-1)^{n}-1}{n}\right) \sin n x\right] \end{array}$$

Step-by-Step Solution

Answer

The Fourier series is derived from the integration of each term, yielding a0, an, and bn coefficients as specified.

Step 1: Calculate the Constant Term a0

Start with the integral for the constant term, which is the average value of the function over one period. Given \( a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx = \frac{1}{\pi} \int_{0}^{\pi} (e^x - 1) \, dx\), solve this integral: \[ \int (e^x - 1) \, dx = e^x - x + C \]\[ a_0 = \frac{1}{\pi} \left[ (e^x - x) \right]_0^\pi = \frac{1}{\pi} (e^\pi - \pi - 1) \].

Step 2: Calculate Cosine Coefficients an

For \( a_n \), express as: \( a_n = \frac{1}{\pi} \int_{0}^{\pi} (e^x - 1) \cos(nx) \, dx\). Use integration by parts for each portion of the integral. This yields: \[ a_n = \frac{[e^\pi (-1)^n - 1]}{\pi(1 + n^2)} \].

Step 3: Calculate Sine Coefficients bn

For \( b_n \), express as: \( b_n = \frac{1}{\pi} \int_{0}^{\pi} (e^x - 1) \sin(nx) \, dx\). Also use integration by parts, which simplifies to: \[ b_n = \frac{1}{\pi}\left( \frac{n e^\pi (-1)^{n+1}}{1+n^2} + \frac{n}{1+n^2} + \frac{(-1)^n}{n} - \frac{1}{n} \right) \].

Step 4: Formulate the Fourier Series

Combine the results from Steps 1, 2, and 3 to form the entire Fourier series representation of \( f(x) \). The Fourier series is: \[ f(x) = \frac{e^\pi - \pi - 1}{2 \pi} + \sum_{n=1}^{\infty} \left[ \frac{e^\pi (-1)^n - 1}{\pi(1+n^2)} \cos(nx) + \left( \frac{n}{1+n^2} [e^\pi (-1)^{n+1} + 1] + \frac{(-1)^n - 1}{n} \right) \sin(nx) \right] \].

Key Concepts

Integration by Parts

Integration by parts is a fundamental technique used to solve integrals, particularly useful when dealing with products of functions. You might be familiar with it from calculus. The formula is derived from the product rule of differentiation, and it goes like this:\[\int u \, dv = uv - \int v \, du\]

• Choose \( u \) and \( dv \) such that their derivatives/simplifications are easy to handle.
• Compute \( du \) by differentiating \( u \).
• Integrate \( dv \) to find \( v \).
• Apply the integration by parts formula.

In the context of Fourier series calculations, specifically for the cosine and sine coefficients, this technique helps simplify complex integrations. When integrating products of exponential or trigonometric functions, it's a go-to method that not only makes the task feasible but also highlights the interrelationships between different function types.

Cosine Coefficients

Cosine coefficients, represented as \( a_n \), are part of the Fourier series, capturing the even components of the function.These coefficients determine how much of each cosine wave, scaled by \( n \), is needed to reconstruct the function. The formula to find them is:\[a_n = \frac{1}{\pi} \int_{0}^{\pi} f(x) \cos(nx) \, dx\]To compute these coefficients, we typically use integration by parts.Step-by-step process:

• The function \( f(x) \) is integrated over one period times the cosine of \( nx \).
• Integration by parts breaks down complex products into manageable integrals.
• Often involves alternating series components, especially with functions like \( e^x \).

This part of the Fourier series is crucial for capturing the function's overall shape and aligning with the symmetrical parts of the waveform, represented by cosine.

Sine Coefficients

Sine coefficients, denoted as \( b_n \), determine the contribution of sine waves to the reconstructed function within the Fourier series.These coefficients capture the odd components of the function and are computed using:\[b_n = \frac{1}{\pi} \int_{0}^{\pi} f(x) \sin(nx) \, dx\]Using integration by parts is necessary here as well.Soliciting the Sine Components:

• Again, apply integration by parts to handle the \( f(x) \sin(nx) \) product.
• Each \( b_n \) aligns the series with shifts in wave symmetry.
• Occasionally, post-integration simplifications involve alternating terms depending on \( n \).

The sine coefficients are essential for detailed modeling of any asymmetric features in the function. By assigning the correct amplitude to each sine component, these coefficients help craft the overall waveform more accurately.