Question

Consider the function \(u(x, y)=x^{3}-3 x y^{2}\). a. Show that \(u(x, y)\) is harmonic; that is, \(\nabla^{2} u=0\) b. Find its harmonic conjugate, \(v(x, y)\). c. Find a differentiable function, \(f(z)\), for which \(u(x, y)\) is the real part. d. Determine \(f^{\prime}(z)\) for the function in part c. [Use \(f^{\prime}(z)=\frac{\partial_{2}}{\partial x}+i \frac{\partial v}{\partial x}\) and rewrite your answer as a function of \(z .]\)

Short answer

a. The function \(u(x, y) = x^3 - 3xy^2\) is a harmonic function because \(∇^2 u = 0\). b. Its harmonic conjugate is \(v(x, y) = y^3 - 3x^2y + c\). c. The differentiable function is then \(f(z) = z^3 + c\). d. The derivative of this function is \(f^{\prime}(z) = 3z^2\).

Step-by-step solution

Prove \(u(x, y)\) is harmonic

To check if \(u(x, y)\) is harmonic, we calculate the Laplacian, \(∇^2 u\). This is the sum of the second partial derivatives of u with respect to x and y: \[∇^2 u = \frac{∂^2u}{∂x^2} + \frac{∂^2u}{∂y^2}\]

Calculate the Laplacian

Upon calculating, we get: \[∇^2 u = \frac{∂^2u}{∂x^2} + \frac{∂^2u}{∂y^2} = 6x-6x = 0\] as both terms cancel out, proving that \(u(x, y)\) is indeed harmonic.

Find the harmonic conjugate

The harmonic conjugate \(v(x, y)\) needs to satisfy the Cauchy-Riemann equations: \[\frac{∂u}{∂x} = \frac{∂v}{∂y} \quad and \quad -\frac{∂u}{∂y} = \frac{∂v}{∂x}\] Now let's solve these equations.

Calculate the harmonic conjugate

After solving the Cauchy-Riemann equations, the harmonic conjugate is found to be \(v(x, y) = y^3 - 3x^2y + c\), where c is the constant of integration.

Find the differentiable function, \(f(z)\)

To find the complex function \(f(z)\), we combine the original function \(u(x, y)\) and its conjugate \(v(x, y)\) to get: \(f(z) = u(x, y) + iv(x, y) = x^3 - 3xy^2 + i(y^3 - 3x^2y + c)\). This is the intended complex function. Rewriting z as \(x + iy\), we get: \(f(z) = z^3 + c\).

Determine \(f^{\prime}(z)\)

The derivative of \(f(z)\) is found by using the definition: \(f^{\prime}(z)=\frac{∂}{∂x} + i \frac{∂v}{∂x}\). Upon doing the calculus, we get: \(f^{\prime}(z) = 3z^2\).

Key concepts

Harmonic Conjugate

In complex analysis, the concept of a harmonic conjugate comes into play when dealing with harmonic functions, which are solutions to Laplace's equation. A function is called harmonic if it satisfies the property \(abla^2 u = 0\).
The harmonic conjugate helps us form an analytic function from a harmonic one.
To find a harmonic conjugate of a function, such as \(u(x, y)\), we aim to pair it with another function \(v(x, y)\) such that together they satisfy the Cauchy-Riemann equations.
This pair can then be rewritten as a complex function \(f(z) = u(x, y) + iv(x, y)\), where the real part is \(u\) and the imaginary part is \(v\).
These two parts must intertwine to satisfy the conditions allowing \(f(z)\) to be well-defined and differentiable.
For the exercise you encountered, after solving the Cauchy-Riemann equations, the harmonic conjugate was found to be \(v(x, y) = y^3 - 3x^2y + c\). This result not only confirms the conjugacy but also sets the stage for constructing a neat and complete analytical function.

Cauchy-Riemann Equations

The Cauchy-Riemann equations are crucial in complex analysis for establishing the necessary conditions for a function to be differentiable in the complex sense.
They form a bridge between the real-valued functions of two variables \(x\) and \(y\) and their complex representation \(z = x + iy\).
The equations are given by:

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

These help in verifying that a function is complex-differentiable.
Essentially, they ensure that the real and imaginary parts (\(u\) and \(v\)) of a complex function exhibit compatible changes at every point in the domain.
When applied, as in this exercise, they allow for determining the harmonic conjugate of \(u(x, y)\), leading to comprehending how these parts weave into a single analytic function. Without fulfilling these equations, a function cannot be considered analytic at that point.

Complex Analysis

Complex analysis is a fascinating field of mathematics that delves into functions of complex numbers.
It extends the ideas of calculus into the complex plane where functions often behave in more structured and predictable ways compared to their real counterparts.
Key concepts include:

• Analytic functions: Those that are differentiable in the complex sense and satisfy the Cauchy-Riemann equations.
• Holomorphic functions: Another term for functions that are universally analytic throughout a given domain.
• Complex integration and residues: Fundamental tools for solving more intricate problems.

Complex analysis not only eases the manipulation of functions like in the given problem, where the harmonic function transforms to \(f(z) = x^3 - 3xy^2 + i(y^3 - 3x^2y + c)\), but it opens doors to understanding phenomena across physics and engineering contexts.
With functions often rewritten as powers of \(z\), it offers elegant solutions and insights that pure real analysis cannot provide.

Laplacian

The Laplacian is a differential operator denoted as \(abla^2\) and is crucial in various fields like mathematics, physics, and engineering.
It measures how a function diverges from its mean value around a point, effectively "smoothing" out the function.
For the function in your exercise, the Laplacian is calculated as:

• \(abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\)

Finding that \(abla^2 u = 0\) confirms that \(u(x, y)\) is harmonic.
In simpler terms, the function doesn't "bump up" or "sink down" in its surrounding - it remains flat or balanced at every point.
Understanding the Laplacian allows one to ascertain whether transformation to an analytic function is feasibly maintained. This balance is vital when considering how differentials of functions align in physical systems and theoretical models, ensuring continuity and smoothness in modeling real-world scenarios.