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. $$x y$$

Short answer

The function \( xy \) is harmonic, and \( f(z) = xy + i (\frac{y^2}{2} - \frac{x^2}{2}) \). Function \( v(x, y) = \frac{y^2}{2} - \frac{x^2}{2} \) is also harmonic.

Step-by-step solution

Verify that the function is harmonic

To determine if the function is harmonic, it must satisfy Laplace's equation: \[ abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] Given function: \( u(x, y) = xy \)First, compute the second partial derivatives of \(u\):\[ \frac{\partial u}{\partial x} = y \]\[ \frac{\partial u}{\partial y} = x \]Second partial derivatives are:\[ \frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x} (y) = 0 \]\[ \frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial y} (x) = 0 \]Therefore, \[ abla^2 u = 0 + 0 = 0 \]. Hence, the function is harmonic.

Find the corresponding complex function

To find the function \(f(z)\) for which \( u(x, y) = \Re(f(z)) \), express \( f(z) \) where \( z = x + iy \). Let:\[ f(z) = \, u(x, y) + iv(x, y) = \, xy + iv(x, y) \]Take the derivatives with respect to \(x\) and \(y\), and apply the Cauchy-Riemann equations to identify \(v(x, y)\).Cauchy-Riemann equations:\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]From Step 1: \[ \frac{\partial u}{\partial x} = y \quad \text{and} \quad \frac{\partial u}{\partial y} = x \]Thus:\[ \frac{\partial v}{\partial y} = y \quad \text{and} \quad -\frac{\partial v}{\partial x} = x \]Integrate \( \frac{\partial v}{\partial y} \, : \, v = \frac{y^2}{2} + g(x)\)From \( -\frac{\partial v}{\partial x} = x \, :\)\[ - \frac{\partial {\frac{y^2}{2} + g(x)}}{\partial x} = x \]Thus \( g(x) \, = \, -\frac{x^2}{2} \): \, \[ v \, = \, \frac{y^2}{2} - \frac{x^2}{2} \].

Verify that \(~v(x, y)\~) is harmonic

To verify that \( v(x,y) \) is harmonic, check if it satisfies Laplace's equation:\[ abla^2 v = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0 \]First, find \(\frac{\partial^2 v}{\partial x^2} \) and \(\frac{\partial^2 v}{\partial y^2} \): \[ v = \frac{y^2}{2} - \frac{x^2}{2}\]\[ \frac{\partial v}{\partial x} = -x \; and \; \frac{\partial v}{\partial y} = y \]Second partial derivatives are:\[ \frac{\partial^2 v}{\partial x^2} = \frac{\partial (-x)}{\partial x} = -1 \]\[ \frac{\partial^2 v}{\partial y^2} = \frac{\partial y}{\partial y} = 1 \]Add these second partial derivatives: \, \[ \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = -1 + 1 = 0 \]

Key concepts

Laplace's Equation

Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace. It's used to describe the behavior of scalar fields such as electric potential, fluid velocity, or the temperature distribution. The equation is written as: \[ abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. \] If a function satisfies this equation, it is called a harmonic function. Harmonic functions appear in various branches of physics and engineering because they describe steady-state heat distribution, gravitational potential, and many other static conditions. To determine if a function, like \(u(x, y) = xy\), is harmonic, calculate its second partial derivatives with respect to \(x\) and \(y\), and check if their sum equals zero.

Partial Derivatives

Partial derivatives are used to measure how a function changes as its variables change. For a function \(u(x, y)\), the partial derivative with respect to \(x\) is written as \(\frac{\partial u}{\partial x}\), and it shows the rate of change of \(u\) when \(x\) changes, while keeping \(y\) constant. Similarly, \(\frac{\partial u}{\partial y}\) shows the change when \(y\) changes, while \(x\) remains constant. To explore the harmonic nature of \(u(x, y) = xy\), we first find its partial derivatives:

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

Next, we calculate the second partial derivatives:

• \(\frac{\partial^2 u}{\partial x^2} = 0\)
• \(\frac{\partial^2 u}{\partial y^2} = 0\)

These calculations show that the sum of the second partial derivatives is zero, thus \(u(x, y)\) satisfies Laplace's equation.

Cauchy-Riemann Equations

The Cauchy-Riemann equations link the real and imaginary parts of a complex function to ensure they are differentiable. If \(f(z) = u(x, y) + iv(x, y)\) is a complex function, where \(z = x + iy\), then the Cauchy-Riemann equations are: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \text{ and } \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. \] For the given function \(u(x, y) = xy\), the partial derivatives from earlier are:

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

Applying the Cauchy-Riemann equations, we find:

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

Integrating these, we get \(v = \frac{y^2}{2} - \frac{x^2}{2}\). This shows the imaginary part of the complex function, and it's solution to Laplace's equation as well.

Complex Functions

Complex functions are composed of a real part and an imaginary part. They are written as \(f(z) = u(x, y) + iv(x, y)\). In our example, we found the complex function with the initial real part \(u(x, y) = xy\) as: \[ f(z) = xy + i\frac{y^2}{2} - i\frac{x^2}{2}. \] Complex functions are useful in various areas of mathematics and engineering, such as fluid dynamics, electromagnetism, and complex analysis. By using complex numbers (\(z = x + iy\)), we can solve problems more succinctly and see the interplay between two dimensions. Lastly, both the real part \(u\) and the imaginary part \(v\) of a complex function should satisfy Laplace's equation to be considered harmonic.