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.
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.