Question

Derive $$ I(x, \theta)= \begin{cases}\frac{\kappa}{4 \pi} \int_{x}^{0} e^{\frac{\kappa\left(x^{\prime}-x\right)}{\cos \theta}} B\left(x^{\prime}\right) \frac{d x^{\prime}}{-\cos \theta} & (-1<\cos \theta<0) \\ \frac{\kappa}{4 \pi} \int_{-\infty}^{x} e^{\frac{k\left(x^{\prime}-x\right)}{\cos \theta}} B\left(x^{\prime}\right) \frac{d x^{\prime}}{\cos \theta} & (0<\cos \theta<1)\end{cases} $$ and give the physical significance of these relations. (b) Show that \(B(x)\) obeys a linear homogeneous integral equation $$ B(x)=\int_{-\infty}^{0} d x^{\prime} K\left(x-x^{\prime}\right) B\left(x^{\prime}\right) $$ and evaluate the kernel \(K\left(x-x^{\prime}\right)\) in terms of the exponential integral function \(E_{1}(z)=\int_{1}^{\infty}(d t / t) e^{-t z} .\) [Note that this integral equation is essentially of the Wiener-Hopf form (1 \(1-65\) ); its sotution may be found in Morse and Feshbach (M9) Section 8.5.]

Short answer

1. Evaluate integrals based on \(\cos \theta\). 2. Interpret physical significance. 3. Express \(B(x)\) as an integral equation with exponential integral for kernel.

Step-by-step solution

Understanding the given integral form

Identify the two cases for the integral expression of \(I(x, \theta)\) based on the range of \(\cos \theta\). The given integral has different forms depending on whether \(\cos \theta\) is negative or positive.

Case 1 for \(-1 < \cos \theta < 0\)

For \(-1 < \cos \theta < 0\), the integral is: \[ I(x, \theta) = \frac{\kappa}{4 \pi} \int_{x}^{0} e^{\frac{\kappa(x' - x)}{\cos \theta}} B(x') \frac{dx'}{-\cos \theta} \] Pay attention to the negative sign in the denominator, which affects the integration.

Simplifying the first integral

Rewrite the integral by recognizing the integration limits and the effect of the negative sign in the denominator. Change the integration variable if necessary and simplify the exponential expression inside the integral.

Case 2 for \(0 < \cos \theta < 1\)

For \(0 < \cos \theta < 1\), the integral is: \[ I(x, \theta) = \frac{\kappa}{4 \pi} \int_{-\infty}^{x} e^{\frac{\kappa(x' - x)}{\cos \theta}} B(x') \frac{dx'}{\cos \theta} \] Evaluate and simplify this integral by considering the range of integration.

Understanding the physical significance

Interpret the physical meaning of these integrals \(I(x, \theta)\). These expressions can describe the intensity of a physical quantity (such as radiation) at a point \(x\), depending on the contribution over a range influenced by \(\theta\). The variable \(B(x)\) often represents a source term or initial distribution that affects the spread.

Formulate the integral equation for \(B(x)\)

Show that \(B(x)\) satisfies a linear homogeneous integral equation. By definition, \(B(x) = \int_{-\infty}^{0} dx' K(x - x') B(x')\), where \(K(x - x')\) is the kernel to be determined.

Evaluate the kernel \(K(x - x')\) using exponential integral function

Express the kernel function \(K(x - x')\) in terms of the exponential integral function. Recall that \[E_1(z) = \int_{1}^{\infty} \frac{dt}{t} e^{-tz}\] and recognize the relationship to transform the integral equation and determine the specific form of the kernel.

Key concepts

Homogeneous Integral Equation

A **homogeneous integral equation** is a mathematical expression where a function is defined in terms of its integral. The function appears on both sides of the equation, making it a powerful tool in various fields – including physics. In this exercise, we are asked to show that the function \(B(x)\) satisfies a homogeneous integral equation:
\[ B(x) = \int_{-\infty}^{0} K(x - x') B(x') \, dx' \]
Here, the kernel \( K(x - x') \) acts as a weight function that determines how different points \( x' \) contribute to the value of \(B(x)\) .
**Key Characteristics of Homogeneous Integral Equations:**

• They involve an unknown function to be solved.
• The integral range is crucial – in this case, it spans from \( -\infty \) to \( 0 \) .
• The form of the kernel \( K(x - x') \) determines the nature of the solution.

In physics, these equations can model situations where the influence of one part of a system on another must be taken into account over a continuous range.

Exponential Integral Function

The **exponential integral function** is a special function denoted as \(E_1(z)\). It plays a critical role in evaluating complex integrals. For this exercise, we need to express the kernel \( K(x - x') \) using this function. The exponential integral function is defined as:
\[ E_1(z) = \int_{1}^{\infty} \frac{e^{-tz}}{t} \, dt \]
Here's how it helps in our context:

• The function \( E_1(z) \) simplifies the representation of the integral kernel.
• It turns complex integrals into more manageable forms.
• **Usage in Physics:** In physical equations, such functions can simplify the solutions of otherwise daunting integral equations.

For the kernel in our integral equation, recognizing this relationship allows you to transform the integral in a streamlined way, making the solution more tractable.

Physical Significance

Understanding the **physical significance** of the integral equation and its solution is crucial. These equations often appear in physics to describe phenomena where values at every point are interconnected. By solving such equations, we gain insight into how a system evolves or behaves under specific conditions.
For the given integral equation:

• The integral form of \(I(x, \theta)\) describes the intensity of a physical quantity, which could be radiation, heat, or another field.
• The upper and lower bounds of integration reflect different physical situations – either confined (negative cosine) or unbounded (positive cosine) regions.
• **Role of \( B(x) \):** This function typically represents an initial state or source affecting the system's behavior over space.
• Interpreting the forms and solutions in context allows one to predict distributions and changes dynamically.

By employing integral equations, physicists can model and predict the behavior of complex systems accurately, providing deeper insights into the laws governing the natural world.