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. $$\ln \left(x^{2}+y^{2}\right)$$

Short answer

The function \( \ln(x^{2}+y^{2}) \) is harmonic, and \( v(x, y) \) is \( 0 \).

Step-by-step solution

Verify the given function is harmonic

To show that a function is harmonic, confirm it satisfies Laplace's equation. Laplace's equation in two dimensions is: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] First, consider the given function: \[ u(x, y) = \ln(x^2 + y^2) \] Calculate the second partial derivatives with respect to x and y.

Calculate the first partial derivatives

Compute the first partial derivatives of \( u(x,y) \): \[ \frac{\partial u}{\partial x} = \frac{2x}{x^2 + y^2} \] \[ \frac{\partial u}{\partial y} = \frac{2y}{x^2 + y^2} \]

Calculate the second partial derivatives

Next, compute the second partial derivatives: \[ \frac{\partial^2 u}{\partial x^2} = \frac{2(y^2 - x^2)}{(x^2 + y^2)^2} \] \[ \frac{\partial^2 u}{\partial y^2} = \frac{2(x^2 - y^2)}{(x^2 + y^2)^2} \]

Show that the sum of second partial derivatives equals zero

Add the second partial derivatives to verify Laplace's equation: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{2(y^2 - x^2)}{(x^2 + y^2)^2} + \frac{2(x^2 - y^2)}{(x^2 + y^2)^2} = 0 \] Since the sum equals zero, the function \( u(x, y) = \ln(x^2 + y^2) \) is harmonic.

Find a function f(z) whose real part is the given function

Use the complex variable \( z = x + iy \) to express \( f(z) \). Note that: \[ |z|^2 = x^2 + y^2 \] Hence, let: \[ f(z) = \ln|z|^2 \]

Find v(x, y) such that f(z) = u(x, y) + iv(x, y)

Rewrite \( f(z) \) as: \[ f(z) = \ln|z|^2 = \ln(x^2 + y^2) + i \cdot 0 \] Identify: \[ v(x, y) = 0 \]

Verify that v(x, y) satisfies Laplace's equation

Since \( v(x, y) = 0 \), the partial derivatives are: \[ \frac{\partial v}{\partial x} = 0 \] \[ \frac{\partial v}{\partial y} = 0 \] Thus: \[ \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0 + 0 = 0 \] Hence, \( v(x, y) \) also satisfies Laplace's equation.

Key concepts

Laplace's equation

To determine whether a function is harmonic, we need to verify that it satisfies Laplace's equation. In mathematical terms, for a function to be harmonic, the sum of its second partial derivatives with respect to each variable should be zero. This can be expressed as:

\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]
In simpler terms, this means you take the second derivatives of the function with respect to x and y. Then, you add them together. If the sum is zero, the function is harmonic. This is a critical aspect in various fields, such as physics and engineering, because it describes a state of equilibrium.

Partial derivatives

Partial derivatives represent the rate at which a function changes as one particular variable changes, while keeping other variables constant.

To determine if our function \( u(x, y) = \ln(x^2 + y^2) \) is harmonic, we first find its partial derivatives. Compute the first partial derivatives:

\[ \frac{\partial u}{\partial x} = \frac{2x}{x^2 + y^2} \] \[ \frac{\partial u}{\partial y} = \frac{2y}{x^2 + y^2} \]
Next, we calculate the second partial derivatives:
\[ \frac{\partial^2 u}{\partial x^2} = \frac{2(y^2 - x^2)}{(x^2 + y^2)^2} \] \[ \frac{\partial^2 u}{\partial y^2} = \frac{2(x^2 - y^2)}{(x^2 + y^2)^2} \]
By adding these second partial derivatives, we see that:
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]
This confirms that our function is indeed harmonic.

Complex variables

Complex variables extend our understanding of real numbers by adding an imaginary component. A complex number \( z \) can be written as \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part. In this exercise, we used the complex variable \( z \) to find the function \( f(z) \) whose real part is our given function.
We note that the magnitude of a complex number \( z \), denoted by \( |z| \), is given by:

\[ |z|^2 = x^2 + y^2 \]
So we can write:

\[ f(z) = \ln|z|^2 \]
Then:

\[ f(z) = \ln(x^2 + y^2) + i \cdot 0 \]
This shows that the imaginary part of \( f(z) \) is zero, meaning the function \( v(x, y) \) derived from the imaginary part of \( f(z) \) is just zero.

Harmonic functions

Harmonic functions are solutions to Laplace's equation and have significant implications in both theory and application. These functions often represent state of equilibrium or steady state systems in physics and engineering. To verify a harmonic function, we follow through with steps to ensure its second partial derivatives sum to zero.
After verifying that \( u(x, y) = \ln(x^2 + y^2) \) satisfies Laplace's equation, and finding the corresponding function \( f(z) \) in the form of complex variables, we turned our attention to the function \( v(x, y) \).
Since \( v(x, y) = 0 \), it is straightforward to see it satisfies Laplace's equation as well, confirming the function \( v(x, y) \) derived from \( f(z) \) also adheres to the properties of harmonic functions. This entire process demonstrates the deep interconnections between Laplace's equation, harmonic functions, and complex variables.