Problem 6 - Advanced Engineering Mathematics

Question

From Theorem 20.5 \\[ u(x, y)=\frac{y}{\pi} \int_{-\infty}^{\infty} \frac{\cos t}{(x-t)^{2}+y^{2}} d t=\frac{y}{\pi} \int_{-\infty}^{\infty} \frac{\cos (x-s)}{s^{2}+y^{2}} d s \\] letting \(s=x-t .\) But \(\cos (x-s)=\cos x \cos x+\sin x \sin s .\) It follows that \\[ u(x, y)=\frac{y \cos x}{\pi} \int_{-\infty}^{\infty} \frac{\cos s}{s^{2}+y^{2}} d s+\frac{y \sin x}{\pi} \int_{-\infty}^{\infty} \frac{\sin s}{s^{2}+y^{2}} d s=\frac{y \cos x}{\pi}\left(\frac{\pi e^{-y}}{y}\right)=e^{-y} \cos x, \quad y>0 \\]

Step-by-Step Solution

Answer

The solution is \( u(x, y) = e^{-y} \cos x \) for \( y > 0. \)

Step 1: Understanding the Problem Statement

The problem involves finding the function \( u(x, y) \) using an integral transformation, particularly with the given cosine functions. We'll evaluate an integral to simplify the expression for \( u(x, y) \).

Step 2: Transforming the Cosine Function

We start by making a substitution \( s = x-t \), which transforms \( \ \cos(x-s) = \ \cos x \cos s + \sin x \sin s. \ \) This allows us to separate the integral into two parts based on cosine and sine terms.

Step 3: Integrating the Cosine Component

The first part is the integral of the form \( \ \int_{-\infty}^{\infty} \frac{\cos s}{s^2 + y^2} ds. \ \) This integral evaluates to \( \ \frac{\pi e^{-y}}{y} \ \) when solved using complex analysis.

Step 4: Eliminating the Sine Component

The second integral \( \ \int_{-\infty}^{\infty} \frac{\sin s}{s^2 + y^2} ds \ \) evaluates to zero because it is an odd function being integrated over a symmetric interval from negative to positive infinity.

Step 5: Combining the Results

Combine the results of the integrals: \( \ \frac{y \cos x}{\pi}\left(\frac{\pi e^{-y}}{y}\right) = e^{-y} \cos x. \ \) The other term involving sine vanishes. This provides the final simplified result for \( u(x, y) \).

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry, denoted as \( \cos \theta \), where \( \theta \) is an angle. It provides the horizontal coordinate of a point on the unit circle when given an angle. This periodic function exhibits a range of values between -1 and 1, repeating every \( 2\pi \).

• A cosine function is often used in physics and engineering to model wave patterns, including sound and light waves.
• Its mathematical properties make it valuable in various transformations like Fourier transforms.

In this exercise, the cosine function \( \cos t \) is integrated over a complex expression to find \( u(x, y) \). Using trigonometric identities, \( \cos(x-s) \) is expanded, which facilitates separating the integral into more manageable components.

Complex Analysis

Complex analysis is a branch of mathematics that studies functions of complex numbers. Incorporating both real and imaginary components, complex numbers are represented as \( a + bi \), where \( i \) is the square root of -1.

• This field allows for intricate evaluations of integrals that arise in many branches of mathematics and physical sciences.
• Complex analysis enables techniques such as contour integration, which simplifies integration by deforming paths of integration in the complex plane.

In the original exercise, complex analysis is crucial in evaluating the integral \( \int_{-\infty}^{\infty} \frac{\cos s}{s^2 + y^2} ds \). This requires complex analysis techniques to handle the integral of a transformed cosine function across an infinite interval.

Integral Evaluation

Integral evaluation involves calculating the value of integrals, which represent the area under a curve over a certain interval. In calculus, this is a key concept for solving problems related to accumulation and total change.

• There are various techniques in integral evaluation, such as substitution, integration by parts, and using special functions like Laplace Transforms.
• Certain integrals need advanced methods such as complex analysis to be evaluated accurately, especially on infinite intervals.

In this problem, the integral \( \int_{-\infty}^{\infty} \frac{\cos s}{s^2 + y^2} ds \) is evaluated using complex analysis. The complexity of the integral lies in its infinite limits and the nature of the function being integrated.

Odd Function

An odd function is a concept in mathematics where the function \( f(x) \) satisfies the property \( f(-x) = -f(x) \) for all \( x \) in the domain. Graphically, these functions have rotational symmetry about the origin.

• Common examples include \( f(x) = x^3 \) and the sine function, \( \sin x \).
• A key property of odd functions is that their integral over a symmetric interval \([-a, a]\) is zero.

In the exercise, the term \( \int_{-\infty}^{\infty} \frac{\sin s}{s^2 + y^2} ds \) vanishes because \( \sin s \) is an odd function. This integral cancels out over the symmetric bounds of integration, simplifying the solution to the exercise.