Question

Find the mixed Fourier cosine series. $$ f(x)=\left\\{\begin{array}{ll} 1, & 0 \leq x \leq \frac{L}{2} \\ 0, & \frac{L}{2}

Short answer

The mixed Fourier cosine series representation of a given function \(f(x)\) defined on the interval \([0, L]\) is given by: $$f(x) = \sum_{n=0}^{\infty}a_n\cos\left(\frac{n\pi x}{L}\right)$$ where the coefficients \(a_n\) are calculated using the formula: $$a_{n} = \frac{2}{L}\int_{0}^{L}f(x)\cos\left(\frac{n\pi x}{L}\right)dx$$

Step-by-step solution

Determine the coefficients \(a_{n}\)

First, compute the \(a_{n}\) coefficients by plugging the given function into the formula: $$a_{n} = \frac{2}{L}\int_{0}^{L}f(x)\cos\left(\frac{n\pi x}{L}\right)dx$$ The given function is defined piecewise, so we need to split the integral into two parts, corresponding to the intervals where the function is defined differently: $$a_{n} = \frac{2}{L}\left(\int_{0}^{\frac{L}{2}}1\cdot\cos\left(\frac{n\pi x}{L}\right)dx + \int_{\frac{L}{2}}^{L}0\cdot\cos\left(\frac{n\pi x}{L}\right)dx\right)$$ The second integral vanishes because of the zero function. So, we only need to evaluate the first integral for \(a_n\): $$a_{n} = \frac{2}{L}\int_{0}^{\frac{L}{2}}\cos\left(\frac{n\pi x}{L}\right)dx$$

Evaluate the integral and simplify

Next, evaluate the definite integral: $$a_{n} = \frac{2}{L}\left[\frac{L}{n\pi}\sin\left(\frac{n\pi x}{L}\right)\right]_{0}^{\frac{L}{2}}$$ Simplify: $$a_{n} = \frac{2}{n\pi}\left[\sin\left(\frac{n\pi(\frac{L}{2})}{L}\right) - \sin\left(\frac{n\pi(0)}{L}\right)\right]$$ $$a_{n} = \frac{2}{n\pi}\left[\sin\left(\frac{n\pi}{2}\right) - 0\right]$$ $$a_{n} = \frac{2}{n\pi}\sin\left(\frac{n\pi}{2}\right)$$

Write the mixed Fourier cosine series

Finally, write the mixed Fourier cosine series representation of the given function using the coefficients \(a_{n}\): $$f(x) = \sum_{n=0}^{\infty}a_n\cos\left(\frac{n\pi x}{L}\right)$$ Plug in the values of \(a_n\): $$f(x) = \sum_{n=0}^{\infty}\frac{2}{n\pi}\sin\left(\frac{n\pi}{2}\right)\cos\left(\frac{n\pi x}{L}\right)$$ This is the mixed Fourier cosine series representation of the given function \(f(x)\).

Key concepts

Mixed Fourier Series

Mixed Fourier Series combines different types of Fourier series to represent a function. Typically, this involves the use of both sine and cosine terms. In the context of this problem, we specifically deal with a "cosine-only" version to suit the constraints of the function's nature, defined over the interval \([0, L]\).

When creating a mixed Fourier series, we find the Fourier coefficients for each term: sine for odd components and cosine for even components. However, for functions that exhibit certain types of symmetry (like even functions), cosine terms alone can perfectly represent the function, simplifying the series by eliminating sine terms. This diminishes the complexity while retaining accuracy.

Understanding this fundamental approach is essential. It not only reflects a precise mathematical formulation but also shows how different series parts can sketch a full depiction of the original function. The result can be visualized as a summation of these components, providing a powerful tool for various applications in signal processing, heat distribution, and more.

Cosine Series

A cosine series aims to represent a function using only cosine terms. It is especially effective for even functions, where symmetry offers a natural advantage. This exercise requires creating a Fourier cosine series from a function that is partly defined over distinct segments of its domain.

To construct a cosine series for a given function, we calculate the requisite coefficients, known as the Fourier cosine coefficients. For functions defined in segments, we perform integration over these segments to isolate each coefficient.

• Compute the integral over the function's valid domain to find the coefficients.
• Utilize definite integrals bounds matching the where the function's definition changes.
• The formula for the coefficient \(a_n\) focuses these calculations specifically on cosine terms to capture even harmonics.

Performing these integrations results in a series that, when summed, represents the function over its entire domain exclusively with cosine terms. This approach leverages the periodic nature of trigonometric functions to replicate the original function as a superposition of these basis functions.

Piecewise Functions

Piecewise functions are defined by different expressions over various intervals. In this exercise, the function \(f(x)\) takes the value 1 on \([0, \frac{L}{2}]\) and 0 on \((\frac{L}{2}, L)\). Approaching such functions with Fourier techniques requires understanding their distinct segments and integrating accordingly.

Here's how to work with piecewise functions:

• Identify the intervals over which the function's definition changes.
• Evaluate integrals separately for each segment, applying the appropriate expression for the function.
• If the function's value is zero over any interval, the corresponding integral contributions naturally disappear, simplifying calculations.

This methodological breakdown lets us handle transformations segment-wise, ensuring each portion contributes to the overall Fourier representation accurately. Such a breakdown is powerful in applications like electrical engineering and signal processing, where systems are often analyzed in pieces rather than as wholes.