Harmonic Function

Harmonic functions are one of the most important functions in complex analysis, as the study of any function for singularity as well residue we must check the harmonic nature of the function. For any function to be Harmonic, it should satisfy the lapalacian equation i.e., ∇²u = 0.

In this article, we have provided a basic understanding of the concept of Harmonic Function including its definition, examples, as well as properties. Other than this, we will also learn about the steps to identify any harmonic function.

What is Harmonic Function?

A harmonic function is a function which meets two criteria. First, it needs to be smooth, meaning it can be continuously and easily differentiated twice. Second, it must follow a specific rule called Laplace’s equation, expressed as:

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In simpler terms, for a function [u(x, y)] to be harmonic, the sum of its second partial derivatives with respect to x and y must be zero.

Harmonic Function Definition

Any smooth function u(x,y) is said to be harmonic if ∇²u = 0.

Where,

∇² is the laplacian operator i.e., ∂²/∂x² + ∂²/∂y².

In simple words, if any smooth function u(x, y) satisfy the equation u_xx + u_yy = 0, then this function u is harmonic fucntion. Where u_xx and u_yy represent second order partial derivative with respect to x and y respectively.

Examples of Harmonic Function

Some of the common examples of harmonic function are:

• Constant Function: u(x, y) = c
• Holomorphic Function: e^x+iy
  • Real Part of Holomorphic Function: e^x cos y i.e.,  Re[e^x+iy].
  • Real Part of Holomorphic Function: e^x sin y i.e.,  Im[e^x+iy].
• f(x, y) = ln(x²+ y²)

Some other examples with three variable are:

• 1/r
• x/r³
• -ln(r² – z²)
• -ln(r + z)

Where r = x² + y² + z².

What are Conjugate Harmonic Functions?

In situations where you have an analytic function ω(z)=u+iv, you can think of “v” as the conjugate harmonic function of “u” and vice versa. In other words:

If you have an analytic function ω₁(z)=u+iv, then ω₂(z)=−v+iu is also an analytic function.

In this context, u and v are considered harmonic conjugates. This means that these functions are connected in a special way, and when you swap the real and imaginary parts, the resulting function remains analytic.

Properties of Harmonic Functions

Some of the common properties of harmonic functions are:

• if ω(z) = u(x,y) + iv(x,y) is analytic in a region A, then both u and v are harmonic functions in A.
• When u(x,y) is harmonic in a connected region A, then u is the real part of an analytic function ω(z) = u(x,y) + iv(x,y).
• If u and v are the real and imaginary parts of an analytic function, they are considered harmonic conjugates.
• Adding two harmonic functions produces another harmonic function
• Not every pair of arbitrary harmonic functions “u” and “v” are necessarily conjugates unless u+iv forms an analytic function.

How to Identify Harmonic Function?

To identify a harmonic function, you can follow these steps.

Step 1: Understand the Basics

• Harmonic functions are smooth and have continuous second derivatives.
• They satisfy Laplace’s equation:

Step 2: Examine the Function

Consider a function u(x, y), for example, u(x, y) = x² – y²

Step 3: Check Continuity and Differentiability

Ensure that u(x, y) is smooth, meaning it is continuous and has continuous first and second derivatives. In our example, u(x, y) = x² – y² is a polynomial, so it’s smooth everywhere.

Step 4: Verify Laplace’s Equation

Apply Laplace’s equation: \(\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0\).

• Calculate : (-2)
• Calculate : (-2)
• Sum: 2 – 2 = 0

Since the sum is zero, the function u(x, y) = x² – y² satisfies Laplace’s equation, indicating that it is a harmonic function.

Read More,

• Complex Number
• Harmonic Mean
• Harmonic Progression

Solved Examples on Harmonic Function

Example 1: Determine if the function u(x,y) = ln(x² +y²) is harmonic.

Solution:

Calculate the partial derivatives of u.

⇒

⇒

Sum the second partial derivatives.

⇒

Since the sum is zero, u(x, y) = In(x² + y²) is harmonic.

Example 2: Check the harmonic nature of u(x, y) = cos(x) cosh(y).

Solution:

Compute the second partial derivatives of u.

= -cos(x) cosh(y)

= \os(x) cosh(y)

Sum the second partial derivatives:.

The sum is zero, indicating that u(x, y) = cos(x) cosh(y) is a harmonic function.

Practice Problems on Harmonic Functions

Problem 1: Investigate whether the function u(x,y)=x³−3xy²+3x²−3y²+1 is harmonic.

Problem 2: Show that u(x,y)=2x(1−y) is a harmonic function. Determine its harmonic conjugate v(x,y).

Problem 3: Prove that the function u(x,y)=e^x2−y2 cos(2xy) is harmonic. Find the harmonic conjugate v(x,y) of u, considering the ambiguity of a constant.

Harmonic Functions – FAQs

1. What is Harmonic Function?

A harmonic function is a real-valued function whose Laplacian is zero within its domain i.e., It satisfies Laplace’s equation, ∇² f = 0.

2. What is the Rule for Harmonic Function?

Rule for Harmonic Function is that Laplacian of a harmonic function is zero i.e., ∇² f = 0.

3. What is a Harmonic Function Example?

One example of Harmonic Function is f(x, y) = sin (x) cosh (y).

4. What is the Difference between Harmonic and Non Harmonic Function?

Harmonic functions satisfy Laplace’s equation, whereas non-harmonic functions do not satisfy this equation.

5. What is Cauchy-Reimann Equation?

For complex functions, the Cauchy-Riemann equations relate partial derivatives. For f(z) = u(x, y) + iv(x, y), the equations are:

and