Problem 25 - 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 \(f(z)=u+i v .\) \(u(x, y)=x^{2}-y^{2}\)

Step-by-Step Solution

Answer

The harmonic conjugate of \( u(x, y) \) is \( v(x, y) = 2xy \). The analytic function is \( f(z) = z^2 \).

Step 1: Check if the function \( u \) is harmonic

A function is harmonic if it satisfies Laplace's equation: \( abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \). Calculate the partial derivatives:1. \( \frac{\partial u}{\partial x} = 2x \)2. \( \frac{\partial^2 u}{\partial x^2} = 2 \)3. \( \frac{\partial u}{\partial y} = -2y \)4. \( \frac{\partial^2 u}{\partial y^2} = -2 \)Substitute into Laplace's equation:\( abla^2 u = 2 - 2 = 0 \)Since \( abla^2 u = 0 \), \( u(x, y) \) is harmonic.

Step 2: Find the harmonic conjugate \( v \)

The harmonic conjugate \( v(x, y) \) is found by solving the Cauchy-Riemann equations:1. \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \), so \( 2x = \frac{\partial v}{\partial y} \)2. \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \), so \( -2y = -\frac{\partial v}{\partial x} \) or \( \frac{\partial v}{\partial x} = 2y \)Integrate the first equation with respect to \( y \):\( v = \int 2x \, dy = 2xy + C(x) \)Now use the second equation:\( \frac{\partial v}{\partial x} = \frac{d}{dx}(2xy + C(x)) = 2y + \frac{dC}{dx} = 2y \)This implies \( \frac{dC}{dx} = 0 \), so \( C(x) \) is a constant.Thus, \( v(x, y) = 2xy + C \).

Step 3: Form the analytic function \( f(z) \)

The analytic function \( f(z) \) is constructed as:\( f(z) = u(x, y) + iv(x, y) = (x^2 - y^2) + i(2xy) \).This can be expressed as \( f(z) = (x + iy)^2 \), where \( z = x + iy \). Therefore, \( f(z) = z^2 \).

Key Concepts

Laplace's Equation

Laplace's Equation is a second-order partial differential equation that is fundamental in many areas of mathematics and physics. It is given by:\[ abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]This equation states that for a function to be harmonic, the sum of its second partial derivatives with respect to each variable must equal zero.
In simpler terms, a harmonic function does not have any local maxima or minima within the domain it is defined. Imagine the surface created by the function as a perfectly balanced shape, neither peaking nor dipping – that's the harmonic condition. It is often used in potential theory, fluid dynamics, and electromagnetic theory, where steady-state scenarios are modeled.
To verify if a function like \( u(x, y) = x^2 - y^2 \) is harmonic, we calculate its second derivatives and confirm that their sum equals zero, thus satisfying Laplace's Equation and confirming its harmonic nature.

Harmonic Conjugate

A harmonic conjugate is another key concept in complex analysis that pairs up with a harmonic function to form an analytic function. When given a harmonic function \( u(x, y) \), such as \( x^2 - y^2 \), its harmonic conjugate \( v(x, y) \) complements it.
To find a harmonic conjugate, we utilize the Cauchy-Riemann equations that link the partial derivatives of the real function \( u(x, y) \) and the imaginary function \( v(x, y) \), which together create a complex-valued function \( f(z) \). The process involves solving equations like \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).
For our function, the harmonic conjugate \( v(x, y) = 2xy + C \) is derived by integration and ensuring that \( C(x) \) does not affect the imaginary part significantly.

Cauchy-Riemann Equations

The Cauchy-Riemann equations are essential criteria in complex analysis for determining whether a complex function is analytic. Analytic functions, also known as holomorphic functions, are smooth and differentiable across their domains.
These equations are written as:

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

Here, \( u(x, y) \) represents the real part and \( v(x, y) \) the imaginary part. For our function, solving these equations ensured the formation of the harmonic conjugate and verified the conditions for analyticity. By satisfying these equations, we ensure the smooth and differentiable transition from the real plane into the complex plane, allowing the correct formation of a complex function.

Analytic Function

An analytic function is a complex function that is locally given by a convergent power series. This means the function is not only differentiable at every point in its domain but also the derivative itself is continuous.
In our exercise, after establishing that \( u(x, y) = x^2 - y^2 \) is harmonic and finding its harmonic conjugate \( v(x, y) = 2xy \), we can construct the analytic function \( f(z) \) as:\[ f(z) = u(x, y) + iv(x, y) = (x^2 - y^2) + i(2xy) \]This can also be expressed in terms of the complex variable \( z \) as \( f(z) = z^2 \), where \( z = x + iy \).
This formulation is elegant in complex analysis and highlights the strong connection between plane geometry and complex functions. It showcases how harmonic and harmonic conjugate functions pair together seamlessly to form functions that are both stable and highly functional across complex domains.