Question

Find the Fourier cosine series. $$ f(x)=e^{x} ; \quad[0, \pi] $$

Short answer

Question: Find the Fourier cosine series of the function f(x) = e^x, defined on the interval [0, π]. Answer: The Fourier cosine series for the function f(x) = e^x on the interval [0, π] is given by: $$ F(x) = \frac{e^{\pi} - 1}{\pi} + \sum_{n=1}^{\infty} \left[ (e^{\pi} - 1) \frac{2}{n^2\pi} - \frac{2}{\pi} \int_{0}^{\pi} e^x \cos(nx) dx \right] \cos(nx) $$

Step-by-step solution

Find the general formula for coefficients a_n

Using the formula for a_n, we need to calculate the integral for coefficients a_n: $$ a_{n} = \frac{2}{\pi} \int_{0}^{\pi} e^x \cos(nx) dx $$ Now, let's find these coefficients for a_0, a_1, a_2, ...

Calculate the coefficient a_0

For n = 0, let us calculate a_0: $$ a_{0} = \frac{2}{\pi} \int_{0}^{\pi} e^x \cos(0x) dx = \frac{2}{\pi} \int_{0}^{\pi} e^x dx $$ Integrate with respect to x: $$ a_{0} = \frac{2}{\pi} \left[ e^x \right]_{0}^{\pi} = \frac{2}{\pi} (e^{\pi} - 1) $$

Calculate general formula for coefficients a_n for n ≥ 1

For n ≥ 1, let's evaluate the following integral: $$ a_{n} = \frac{2}{\pi} \int_{0}^{\pi} e^x \cos(nx) dx $$ Integration by parts: Let u = e^x and dv = cos(nx) dx. Then, we obtain du = e^x dx and v = 1/n * sin(nx). By applying integration by parts: $$ a_{n} = \frac{2}{\pi} \left[ uv \Big|_0^{\pi} - \int_{0}^{\pi} v \ du \right] $$ Substitute u, dv, du, and v: $$ a_{n} = \frac{2}{\pi} \left[ \frac{e^x}{n} \sin(nx) \Big|_0^{\pi} - \int_{0}^{\pi} \frac{\sin(nx)}{n} \ e^x dx \right] $$ The term e^x * sin(nx) is zero for both x=0 and x=π, so the first term will be zero: $$ a_{n} = - \frac{2}{\pi} \int_{0}^{\pi} \frac{\sin(nx)}{n} \ e^x dx $$ Now, we need to integrate this expression again by parts. Let u = e^x and dv = -sin(nx)/n dx. Then we get du = e^x dx and v = -cos(nx)/n^2. By applying integration by parts again: $$ a_{n} = \frac{2}{\pi} \left[ uv \Big|_0^{\pi} - \int_{0}^{\pi} v \ du \right] $$ Substitute u, dv, du, and v: $$ a_{n} = \frac{2}{\pi} \left[\frac{-e^x}{n^2} \cos(nx) \Big|_0^{\pi} + \int_{0}^{\pi} \frac{\cos(nx)}{n^2} \ e^x dx \right] $$ Now, evaluate the boundary terms and simplify: $$ a_{n} = \frac{2}{n^2\pi} \left[ (e^{\pi} - 1) - n^2 \int_{0}^{\pi} e^x \cos(nx) dx \right] $$ So, $$ a_{n} = (e^{\pi} - 1) \frac{2}{n^2\pi} - \frac{2}{\pi} \int_{0}^{\pi} e^x \cos(nx) dx $$

Write the Fourier cosine series

Combining the results of previous steps, we can write the Fourier cosine series for f(x) = e^x as follows: $$ F(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_{n} \cos(nx) = \frac{e^{\pi} - 1}{\pi} + \sum_{n=1}^{\infty} \left[ (e^{\pi} - 1) \frac{2}{n^2\pi} - \frac{2}{\pi} \int_{0}^{\pi} e^x \cos(nx) dx \right] \cos(nx) $$ This is the Fourier cosine series for the given function f(x) = e^x on the interval [0, π].

Key concepts

Cosine Series

A cosine series is a special type of Fourier series used to represent a periodic function using only cosine terms. This is particularly useful when dealing with even functions, which are symmetric around the y-axis. In a cosine series, we calculate coefficients that help us write the original function as an infinite sum of cosine terms.
The general form of a Fourier cosine series for a function \( f(x) \) defined on the interval \([0, L]\) is given by:

• \(F(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n \pi x}{L}\right)\)

Here, \( a_0, a_1, a_2, \ldots \) are the coefficients to be determined, and \( L \) is the length of the interval.
The series attempts to approximate the function as closely as possible within the interval.

Integration by Parts

Integration by parts is a technique used to integrate products of functions. This method derives from the product rule of differentiation and helps in breaking down complex integrals into more manageable parts. The formula for integration by parts is:

• \(\int u \, dv = uv - \int v \, du\)

When using this technique, you choose one part of the integrand to differentiate (\( u \)) and the other to integrate (\( dv \)). For our problem, when finding the coefficients of the cosine series, the expression we need to integrate involves a product of an exponential function and a trigonometric function.
This requires applying integration by parts multiple times to simplify the original integral and solve for the desired coefficients. It's like peeling an onion, where each layer makes the problem simpler.

Coefficient Calculation

Calculating coefficients in a Fourier series is a crucial step, as these coefficients determine how well the series approximates the given function. For the cosine series, these coefficients are denoted as \( a_n \). Here’s a brief on how they are obtained:

• \( a_0 = \frac{2}{\pi} \int_0^\pi f(x) \, dx \)
• \( a_n = \frac{2}{\pi} \int_0^\pi f(x) \cos(nx) \, dx \) for \( n \geq 1 \)

For the given function \( f(x) = e^x \), you integrate it multiplied by cosine terms over the interval \([0, \pi]\). These integrals usually require sophisticated techniques like integration by parts, as direct integration isn’t straightforward. Calculating these coefficients correctly ensures that we can properly construct our cosine series for the function.

Boundary Value Problems

Boundary value problems often arise in disciplines such as physics and engineering, where you need to find a function that satisfies a differential equation and specific conditions at the boundaries of the domain. These problems naturally lead to the use of Fourier series, including cosine and sine series.
In our context, we use a Fourier cosine series to solve a boundary value problem, where the function \( f(x) \) is defined over a domain \([0, \pi]\). This approach is fitting because cosine functions naturally satisfy many boundary conditions by being zero or having simple values at the boundaries of the interval.

• In other words, the cosine series aligns with the constraints imposed by boundary conditions, making it a powerful tool in solving such problems.

This method provides a systematic way to tackle the problem while adhering to the boundary conditions.