Harmonic Functions

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

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.