Problem 11 - Advanced Engineering Mathematics

Question

Represent the given function by an appropriate cosine or sine integral. $$ f(x)=e^{-|x|} \sin x $$

Step-by-Step Solution

Answer

The function \( f(x) = e^{-|x|} \sin x \) is best represented by its sine transform due to its odd symmetry.

Step 1: Recognize the Function Type

First, observe that the function \[ f(x) = e^{-|x|} \ x \sin(x) \]is an even function multiplied by an odd function, making \[ f(x) \] odd. This hints that the sine integral representation is more appropriate.

Step 2: Break Down the Function

Recognize that the function can be expressed as a combination \[ f(x) = e^{-x} \sin(x) \] for \( x \geq 0 \) and \[ f(x) = -e^{x} \sin(x) \] for \( x \leq 0 \).

Step 3: Sine Transform Formulation

Use the sine transform to represent the function. The sine transform for a function \( f(x) \) is given by:\[ \mathcal{F}_s[f](u) = \int_{-\infty}^{\infty} f(x) \sin(2\pi u x) \, dx \]However, due to the nature of the function, the integration over the negative and positive halves are symmetrical and cancel in partial cases, simplifying the process.

Step 4: Calculate the Integral

Calculate the Fourier sine transform:\[ \mathcal{F}_s[f](u) = \int_{0}^{\infty} e^{-x} \sin(x) \sin(2\pi u x) \, dx \]This integral can be simplified using trigonometric identities and possibly contour integration in an advanced context if needed, commonly leading to tabular integral results.

Key Concepts

Sine Integral

When dealing with transforms of odd functions, the sine integral plays a significant role. In the given problem, the goal is to use a sine integral to represent the function. This involves recognizing that the function, although initially complex in form, can be simplified for calculation purposes. The sine transform specifically focuses on the odd components of a function, capturing its symmetry around the origin. The integral used is generally of the form:

• \( \mathcal{F}_s[f](u) = \int_{0}^{\infty} f(x) \sin(2\pi u x) \, dx \)

This highlights the product of the function and the sine term. For the function \( f(x) = e^{-|x|} \sin(x) \), it allows us to leverage its properties, simplifying the integration process by recognizing patterns and symmetries.

Odd Function

Odd functions are symmetrical around the origin, meaning that \( f(-x) = -f(x) \). In this exercise, recognizing the function \( f(x) = e^{-|x|} \sin(x) \) as odd was a crucial step. Odd functions display a specific waveform symmetry and often lend themselves well to sine transforms.Understanding this symmetry is key, as it assists in breaking down the function more effectively for analysis. Observing that multiplying an even function such as \( e^{-|x|} \) with an odd one like \( \sin(x) \) results in another odd function is pivotal to determining the appropriate transform method. This reflection about the y-axis and the resulting nature of odd functions simplifies many integral calculations by making certain integrals vanish, thus reducing computational effort.

Trigonometric Identities

Trigonometric identities are essential tools in simplifying and solving integral expressions. When dealing with the given function, using identities like these can simplify complex products into more manageable forms.In particular, the identity:

• \( \sin(A)\sin(B) = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \)

is useful in our scenario. By applying such identities, we can rewrite the integrals in a form that is easier to evaluate. For example, using it allows breaking down the product \( \sin(x)\sin(2\pi u x) \) into a combination of cosine terms, which can then be integrated more straightforwardly. This simplification is crucial in both manual calculations and computer-based numerical integration.

Contour Integration

Contour integration is a technique reserved for more advanced applications, helping evaluate integrals in the complex plane. While not always necessary, it provides an alternative pathway for evaluating certain definite integrals. For this exercise, contour integration could be considered if direct computation becomes intractable. It involves creatively choosing a path (or contour) in the complex plane where the integral's value can be more easily computed. Once computed over this path, the result can relate back to the original real-valued integral. Its utility lies in how complex analysis can simplify real integrals that are otherwise difficult to evaluate directly, often leveraging the elegant residues of poles within these contours.