Problem 28 - Advanced Engineering Mathematics

Question

In Problems 23-28, verify that the given function \(u\) is harmonic. Find \(v\), the harmonic conjugate function of \(u\). Form the corresponding analytic function \(f(z)=u+i v\). $$ u(x, y)=e^{x}(x \cos y-y \sin y) $$

Step-by-Step Solution

Answer

u is harmonic if \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\).

Step 1: Define Harmonic Functions

A function \(u(x, y)\) is harmonic if it satisfies Laplace's equation, \(abla^2 u = 0\), where \(abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\).

Key Concepts

Laplace's Equation

Laplace's equation is a fundamental concept in mathematics and physics, especially when dealing with potential fields like electrostatic or gravitational fields. In mathematics, particularly in complex analysis, a function is said to be harmonic if it satisfies this equation. A harmonic function, in two dimensions, must satisfy \[abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\]This essentially means the function is twice differentiable and its Laplacian is zero, indicating that the value at any given point is the average of its values around that point.
Harmonic functions naturally arise as solutions to problems where the Laplace equation is involved, like in fluid dynamics and electromagnetics.
For instance, to verify if a function \(u(x, y)\) is harmonic, we need to ensure that the sum of its second derivatives with respect to \(x\) and \(y\) equals zero. This was part of the process shown in the given exercise, where the function \(u(x, y) = e^{x}(x \cos y - y \sin y)\) was checked using this criterion.

Harmonic Conjugate

Once a function is identified as harmonic, the next step often involves finding its harmonic conjugate. A harmonic conjugate \(v(x, y)\) of a harmonic function \(u(x, y)\) is another function such that the expression \(u + iv\) can be considered an analytic function.
To find the harmonic conjugate, we use the Cauchy-Riemann equations, which link the partial derivatives of \(u\) and \(v\):

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

These equations are crucial because they ensure that \(u\) and \(v\) combine together to form an analytic function. In essence, if \(v\) is correctly chosen or calculated using these equations, \(u + iv\) describes a complex function that is both continuous and differentiable in the complex plane. Finding \(v\) typically requires integration of the derivatives, ensuring that the relationships from the Cauchy-Riemann equations are preserved.

Analytic Functions

Analytic functions are a class of functions that are not only continuous but also differentiable at every point in their domain. In the realm of complex numbers, this differentiability is significantly stronger than in real analysis.
For a function to be analytic, it must satisfy the Cauchy-Riemann equations, thereby allowing it to be expressed as \(f(z) = u(x, y) + i v(x, y)\), where \(u\) and \(v\) are harmonic conjugates.
Analytic functions often exhibit very regular properties, such as having a power series representation around any point within their domain of definition.

• They preserve angles and the shapes of infinitesimally small figures.
• They have derivatives of all orders and can be differentiated term by term.

In our problem, constructing \(f(z) = u(x, y) + iv(x, y)\) once \(v(x, y)\) is determined, produces an analytic function. This representation not only smoothens the interaction between the real and imaginary parts but also seamlessly connects the theory of functions of complex variables to practical, real-world situations.