Harmonic Functions

Harmonic functions occur regularly and play an essential role in maths and other domains like physics and engineering. In complex analysis, harmonic functions are called the solutions of the Laplace equation. Every harmonic function is the real part of a holomorphic function in an associated domain. In this article, you will learn the definition of harmonic function, along with some fundamental properties.

Before learning about harmonic functions, let’s recall the definition of the Laplace equation.

An equation having the second-order partial derivatives of the form

Also, check:Laplace Transform

What is Harmonic Function?

A function u(x, y) is known as harmonic function when it is twice continuously differentiable and also satisfies the below partial differential equation, i.e., the Laplace equation:

∇²u = u_xx + u_yy = 0.

Or

That means a function is called a harmonic function if it satisfies Laplace’s equation.

Conjugate Harmonic Function

If f(z) = u + iv is an analytic function, then “v” is the conjugate harmonic of “u” and vice versa.

Alternative definition:

If f(z) = u + iv is an analytic function, then so is f(z) = −v + iu such that u and v are harmonic conjugates.

Properties of Harmonic Functions in Complex Analysis

1. If f(z) = u(x, y) + iv(x, y) is analytic on a region A then both u and v are harmonic functions on A.
2. If u(x, y) is harmonic on a connected region A, then u is the real part of an analytic function f(z) = u(x, y) + iv(x, y).
3. If u and v are the real and imaginary parts of an analytic function, then we say u and v are harmonic conjugates.
4. The sum of two harmonic functions is a harmonic function.
5. An arbitrary pair of harmonic functions “u” and “v” need not be conjugated unless u + iv is an analytic function.

Finding Harmonic Function

Let’s learn how to find the harmonic function or verify whether the given function is harmonic with the help of an example given below.

Example:

Show that the function u(x, y) = sinh x cos y is harmonic. Find the corresponding conjugate harmonic function, i.e, v(x, y).

Solution:

Given function: u(x, y) = sinh x cos y

∂u/∂x = ∂/∂x (sinh x cos y) = cosh x cos y

And

∂²u/∂x² = ∂/∂x (cosh x cos y) = sinh x cos y

Now,

∂u/∂y = ∂/∂y (sinh x cos y) = sinh x (-sin y) = -sinh x sin y

And

∂²u/∂y² = ∂/∂y (-sinh x sin y) = -sinh x cos y

Thus, (∂²u/∂x²) + (∂²u/∂y²) = sinh x cos y + (-sinh x cos y) = 0

That means u(x, y) is harmonic function.

From the Cauchy Riemann equations, we can express as:

∂u/∂x = ∂v/∂y

As we know, ∂u/∂x = cosh x cos y

Now, to find v, we need to integrate cosh x cosy with respect to dy.

i.e. ∫ cosh x cos y dy

= cosh x ∫ cos y dy

= cosh x sin y + C

Therefore, v(x, y) = cosh x sin y + C

Practice Problems

1. Show that the function v(x, y) = arg(z), z ≠ 0 is harmonic. Find the corresponding conjugate harmonic function, i.e., u(x, y).
2. Show that the function u(x,y)=e^(x²−y²) cos⁡(2xy)is harmonic. Find the harmonic conjugate v of u, up to the addition of a constant.
3. Show that u(x, y) is a harmonic function and find its harmonic conjugate when u(x, y) = 2x(1 − y).
4. Check whether the function u = x³ − 3xy² + 3x² − 3y² + 1 is harmonic.