Problem 7 - Advanced Engineering Mathematics

Question

Represent the given function by an appropriate cosine or sine integral. $$ f(x)=\left\\{\begin{array}{lr} 0, & x<-1 \\ -5, & -11 \end{array}\right. $$

Step-by-Step Solution

Answer

The function can be represented by the sine integral \( f(x) = \int_{-\infty}^{\infty} 10\left(\frac{1 - \cos(k)}{k}\right) \sin(kx) \, dk \).

Step 1: Analyze the Function

The function \( f(x) \) given is a piecewise function with values 0 outside the interval \(-1 < x < 1\), and takes the values \(-5\) for \(-1 < x < 0\) and \(5\) for \(0 < x < 1\). This setup suggests a rectified function defined over the interval \(-1, 1\).

Step 2: Identify Symmetry

The function \( f(x) \) has odd symmetry, as \( f(-x) = -f(x) \). Odd symmetry suggests representing it using a sine integral, which naturally handles odd functions better compared to cosine integrals.

Step 3: Determine the Sine Integral Representation

Given the function's symmetry, we use the sine form of the Fourier integral representation: \[ f(x) = \int_{-\infty}^{\infty} F_s(k) \sin(kx) \, dk\]where \( F_s(k) \) represents the sine transform of \( f(x) \).

Step 4: Find the Sine Transform \( F_s(k) \)

The sine transform \( F_s(k) \) is given by the integral: \[ F_s(k) = \int_{-\infty}^{\infty} f(x) \sin(kx) \, dx\]However, it simplifies to:\[ F_s(k) = \int_{-1}^{0} -5 \sin(kx) \, dx + \int_{0}^{1} 5 \sin(kx) \, dx\] Given the symmetry and piecewise definition of \( f(x) \).

Step 5: Solve the Integrals

Compute the two integrals for \( F_s(k) \): \[ \int_{-1}^{0} -5 \sin(kx) \, dx = 5\left[\frac{\cos(kx)}{k}\right]_{-1}^{0}\] \[ = 5\left(\frac{\cos(0)}{k} - \frac{\cos(-k)}{k}\right) = 5\left(\frac{1 - \cos(k)}{k}\right)\] Similarly, for the second integral:\[ \int_{0}^{1} 5 \sin(kx) \, dx = 5\left[\frac{-\cos(kx)}{k}\right]_{0}^{1}\] \[ = 5\left(\frac{-\cos(k)}{k} - \frac{-1}{k}\right) = 5\left(\frac{1 - \cos(k)}{k}\right)\]Adding these contributions gives the transform:\[ F_s(k) = 10\left(\frac{1 - \cos(k)}{k}\right)\]

Step 6: Substitute Back into Integral

Substitute \( F_s(k) \) back into the sine integral representation:\[ f(x) = \int_{-\infty}^{\infty} 10\left(\frac{1 - \cos(k)}{k}\right) \sin(kx) \, dk\]This represents the original function \( f(x) \) as a cosine/sine integral.

Key Concepts

Piecewise Function

A piecewise function is a function that has different expressions or rules for different intervals of the domain. It is defined by multiple sub-functions, each applying to a specific interval or condition.
In the given problem, we have a piecewise function:

• From \( x < -1 \), the function value is 0.
• When \( -1 < x < 0 \), it takes the value -5.
• For the interval \( 0 < x < 1 \), the function jumps to 5.
• Finally, for \( x > 1 \), it returns to 0.

Such functions are common in mathematical modeling to describe situations where a rule changes based on the input variable. Understanding the domain-specific rules is crucial for rewriting the function in integral form or for further analytical work.

Odd Symmetry

Odd symmetry is a property of functions where \( f(-x) = -f(x) \) for all values of \( x \). This means the function has a reflectional symmetry about the origin.
In our piecewise function, this symmetry is clear since:

• When \( x = 0 \), the function is continuous.
• For reversed input values like \( x = -0.5 \) and \( x = 0.5 \), we have the outputs -5 and 5, respectively.

Odd functions are inherently compatible with the Fourier sine transform. That is because the sine function itself is an odd function. Thus, it provides a natural and efficient tool for representing such symmetrically structured outputs.
Recognizing odd symmetry helps to decide if a sine integral is suitable for expressing a function.

Sine Integral Representation

The sine integral representation is a method of expressing functions as an integral involving the sine function. This approach is particularly useful for functions with odd symmetry.
The general form is: \[ f(x) = \int_{-\infty}^{\infty} F_s(k) \sin(kx) \, dk \]where \( F_s(k) \) is the sine transform of \( f(x) \). Calculating \( F_s(k) \) requires integrating the function multiplied by \( \sin(kx) \) over the domain. In our example, we broke it down into manageable intervals based on the piecewise definition.
Once the sine transform \( F_s(k) \) is determined, it is plugged back into the integral to fully represent the original function globally.

Fourier Transform

A Fourier transform is a mathematical operation that transforms a function from its original domain (often time or position) into the frequency domain. Unlike the direct computation of functions, it allows us to study the frequency components of a signal or function.
There are different types of Fourier transforms, including the sine and cosine transforms. Each is specialized for functions with specific symmetries (odd for sine, even for cosine).

• In our step-by-step solution, we used the Fourier sine transform.
• This is due to the odd symmetry in the given piecewise function.

Calculating a Fourier sine transform involves integrating the product of the function and a sine wave over all space, giving us a way to describe the function as a sum of sine waves. Understanding the Fourier transform concepts and how it connects to symmetry helps in choosing the appropriate integral representation for various functions.