Problem 24 - Advanced Engineering Mathematics

Question

Verify that the given function \(u\) is harmonic. Find \(v\), the harmonic conjugate function of \(u\). Form the corresponding analytic function \(f(z)=u+i v .\) \(u(x, y)=2 x-2 x y\)

Step-by-Step Solution

Answer

The analytic function is \( f(z) = (2x - 2xy) + i(2y - y^2 - x^2 + C) \).

Step 1: Verify Harmonic Condition

A function is harmonic if it satisfies Laplace's equation, \( abla^2 u = 0 \). For \( u(x,y) = 2x - 2xy \), calculate the second partial derivatives.\\[\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x}(2-2y) = 0\]\\[\frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial y}(-2x) = 0\]\Since \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 + 0 = 0 \), \( u(x, y) \) satisfies Laplace's equation, so it is harmonic.

Step 2: Find Harmonic Conjugate

For a harmonic function \( u(x,y) \), the harmonic conjugate \( v(x,y) \) satisfies the Cauchy-Riemann equations: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).Calculate \( \frac{\partial u}{\partial x} = 2 - 2y \) and \( \frac{\partial u}{\partial y} = -2x \).1. From \( \frac{\partial v}{\partial y} = 2 - 2y \): Integrate with respect to \( y \) to find \( v \): \[ v(x,y) = 2y - y^2 + g(x) \] \2. From \( \frac{\partial v}{\partial x} = -2x \): \[ \frac{\partial}{\partial x}(2y - y^2 + g(x)) = g'(x) = -2x \] \ Integrate to find \( g(x) \): \ \[ g(x) = - x^2 + C \] \So, \( v(x,y) = 2y - y^2 - x^2 + C \).

Step 3: Form the Analytic Function

The analytic function \( f(z) \) is formed by combining the harmonic function \( u(x,y) \) and its conjugate \( v(x,y) \):\[ f(z) = u(x,y) + iv(x,y) = (2x - 2xy) + i(2y - y^2 - x^2 + C) \].

Key Concepts

Laplace's equation

Laplace's equation is a second-order partial differential equation essential in mathematical physics. It is represented as \( abla^2 u = 0 \), meaning the sum of the second partial derivatives of a function \( u \) with respect to the variables is zero. This equation characterizes harmonic functions, which are solutions to this equation.

For a function \( u(x, y) \) to be harmonic, it must satisfy this equation in the given domain. In the context of our exercise, the function \( u(x, y) = 2x - 2xy \) was checked against Laplace's equation.

By calculating the second partial derivatives, \( \frac{\partial^2 u}{\partial x^2} = 0 \) and \( \frac{\partial^2 u}{\partial y^2} = 0 \), it was verified that their sum is zero. Hence, \( u(x, y) \) is indeed a harmonic function because it fulfills \( abla^2 u = 0 \).

Harmonic functions have many applications in physics, such as in the modeling of electrostatic potential and fluid dynamics, as they often represent equilibrium states. Learning about Laplace's equation is crucial for understanding complex occurred phenomena in these fields.

Harmonic Conjugate

A harmonic conjugate is another function that pairs with a given harmonic function to form an analytic function. In this context, when a function \( u(x, y) \) is harmonic, it pairs with its conjugate \( v(x, y) \) to create the complex function \( f(z) = u(x, y) + iv(x, y) \). This means \( f(z) \) becomes analytic at that point.

To find the harmonic conjugate, we use the Cauchy-Riemann equations, which must be satisfied by \( u \) and \( v \):

• \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)
• \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

In the exercise, \( \frac{\partial u}{\partial x} = 2 - 2y \) and \( \frac{\partial u}{\partial y} = -2x \) were calculated, leading to the conclusion that \( v(x, y) = 2y - y^2 - x^2 + C \).

This procedure of finding \( v \) complements \( u \), fulfilling its role as a harmonic conjugate, which is key for forming the analytic function and understanding complex mapping.

Cauchy-Riemann equations

The Cauchy-Riemann equations are foundational in complex analysis, establishing the conditions under which a complex function is analytic. They interrelate the partial derivatives of two real-valued functions, \( u(x, y) \) and \( v(x, y) \), which together form a complex number, \( z = x + iy \). These equations boil down to:

• \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)
• \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

These must be satisfied for \( u \) and \( v \) to form a valid analytic function \( f(z) = u(x, y) + iv(x, y) \).

In the exercise, these equations were employed to verify the harmonic nature of \( u \) and to find \( v \), its conjugate, ensuring \( f(z) \) is analytic. By satisfying these equations, you confirm that \( f(z) \) behaves nicely, particularly in the domain of complex analysis, enabling smooth, differentiable behavior.

Understanding how to apply Cauchy-Riemann is crucial, not just for theoretical exploration but also for practical applications, such as solving fluid dynamics problems and analyzing electromagnetic fields.