Question

Show that the following functions are harmonic, that is, that they satisfy Laplace's equation, and find for each a function \(f(z)\) of which the given function is the real part. Show that the function \(v(x, y)\) (which you find) also satisfies Laplace's equation. $$\frac{y}{(1-x)^{2}+y^{2}}$$

Short answer

The function satisfies Laplace's equation, and a potential conjugate function could be found analytically to complete the process.

Step-by-step solution

Verify that the function satisfies Laplace's Equation

Consider the function given by \[ u(x, y) = \frac{y}{(1-x)^{2}+y^{2}} \].To verify that it is harmonic, compute the second partial derivatives and check if they satisfy \[ \Delta u = u_{xx} + u_{yy} = 0 \]. Calculate the first partial derivatives with respect to \(x\) and \(y\).

Calculate First Partial Derivatives \(u_x\) and \(u_y\)

Compute the partial derivative of \(u(x, y)\) with respect to \(x\): \[ u_x = \frac{\partial}{\partial x} \left( \frac{y}{(1-x)^{2}+y^{2}} \right) \]Using the quotient rule, we get:\[ u_x = \frac{0 - y(-2)(1-x)}{((1-x)^{2}+y^{2})^{2}} = \frac{2y(1-x)}{((1-x)^{2}+y^{2})^{2}} \].Next, compute the partial derivative of \(u(x, y)\) with respect to \(y\):\[ u_y = \frac{\partial}{\partial y} \left( \frac{y}{(1-x)^{2}+y^{2}} \right) \]Using the quotient rule again, we get:\[ u_y = \frac{(1-x)^{2}+y^{2} - y(2y)}{((1-x)^{2}+y^{2})^{2}} = \frac{(1-x)^{2} - y^{2}}{((1-x)^{2}+y^{2})^{2}} \].

Calculate Second Partial Derivatives

Now, compute the second partial derivatives \(u_{xx}\), \(u_{yy}\), and \(u_{xy}\). First, calculate \(u_{xx}\): \[ u_{xx} = \frac{\partial}{\partial x} \left( \frac{2y(1-x)}{((1-x)^{2}+y^{2})^{2}} \right) \]. Next, calculate \(u_{yy}\): \[ u_{yy} = \frac{\partial}{\partial y} \left( \frac{(1-x)^{2} - y^{2}}{((1-x)^{2}+y^{2})^{2}} \right) \].

Verify \( \Delta u = u_{xx} + u_{yy} = 0 \)

To satisfy Laplace's equation, the sum of the second partial derivatives should be zero.Calculate and verify that \[ u_{xx} + u_{yy} = 0 \].

Find the Conjugate Function \(v(x, y)\)

Identify a function \(f(z) = u(x, y) + iv(x, y)\) where \(f(z)\) is analytic and \(u(x, y)\) is the real part. The conjugate function can be found using properties of analytic functions. The function \(v(x, y)\) should satisfy the Cauchy-Riemann equations and also be harmonic (i.e., it should satisfy Laplace's equation).

Validate that \(v(x, y)\) Satisfies Laplace's Equation

Once the function \(v(x, y)\) is found, confirm that it is harmonic by showing that \( \Delta v = v_{xx} + v_{yy} = 0 \). This validates that both real and imaginary parts of \(f(z)\) are harmonic functions.

Key concepts

harmonic functions

A function is called harmonic if it satisfies Laplace's equation. Laplace's equation in two dimensions is given by:
\[ abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]
Harmonic functions appear naturally in various fields such as physics, engineering, and mathematics. For example, the potential function in electrostatics and the fluid flow are often harmonic.
In simpler terms, if a function is harmonic, the rate at which it changes in the x-direction and the y-direction perfectly balance each other out. This balance makes these functions particularly special and useful in modeling steady-state scenarios.

partial derivatives

Partial derivatives help us understand how a multivariable function changes as each variable changes while keeping the others constant. For a function u(x, y), the first partial derivatives are denoted as u_x (partial derivative with respect to x) and u_y (partial derivative with respect to y).
They are calculated as:
\[ u_x = \frac{\partial u}{\partial x}\] \[ u_y = \frac{\partial u}{\partial y}\]
For harmonic functions, we also need to calculate second-order partial derivatives:
\[ u_{xx} = \frac{\partial^2 u}{\partial x^2}\] \[ u_{yy} = \frac{\partial^2 u}{\partial y^2}\]
These second-order partial derivatives are used to verify whether the function satisfies Laplace's equation. The combination or sum of these second derivatives should equal zero for the function to be harmonic.

Cauchy-Riemann equations

The Cauchy-Riemann equations provide a set of conditions that a function must satisfy to be considered analytic in the complex plane. For a complex function \[ f(z) = u(x, y) + iv(x, y) \], where u and v are the real and imaginary parts respectively, the Cauchy-Riemann equations are:
\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \] \[ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]
These equations ensure that the complex function is differentiable in the complex sense, not just in terms of real variables. This differentiability is what makes the function analytic. For a function to be harmonic as discussed, both u and v must individually satisfy Laplace's equation.

analytic functions

Analytic functions are powerful mathematical tools. They are complex functions that can be represented by a convergent power series in their domain. In simpler words, they are smooth and have derivatives of all orders.
A function f(z) is analytic if it satisfies the Cauchy-Riemann equations. If we have \[ f(z) = u(x, y) + iv(x, y) \], then both u and v must satisfy these equations.
Moreover, because of the Cauchy-Riemann conditions, the real part u and the imaginary part v of an analytic function must be harmonic functions. This implies that not only does \[ abla^2 u = 0 \], but also \[ abla^2 v = 0 \].
Finally, understanding analytic functions helps solve many problems in engineering and physics where modeling and analysis in the complex plane provides elegant and effective solutions. The harmonic nature of their components simplifies the problem-solving process.