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., ∇ 2 u = 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.
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.
Any smooth function u(x,y) is said to be harmonic if ∇ 2 u = 0.
Where,
∇ 2 is the laplacian operator i.e., ∂ 2 /∂x 2 + ∂ 2 /∂y 2 .
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.
Some of the common examples of harmonic function are:
Some other examples with three variable are:
Where r = x 2 + y 2 + z 2 .
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 ω 1 (z)=u+iv, then ω 2 (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.
Some of the common properties of harmonic functions are:
To identify a harmonic function, you can follow these steps.
Step 1: Understand the Basics
Step 2: Examine the Function
Consider a function u(x, y), for example, u(x, y) = x 2 – y 2
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) = x 2 – y 2 is a polynomial, so it’s smooth everywhere.
Step 4: Verify Laplace’s Equation
Since the sum is zero, the function u(x, y) = x 2 – y 2 satisfies Laplace’s equation, indicating that it is a harmonic function.
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Example 1: Determine if the function u(x,y) = ln(x 2 +y 2 ) is harmonic.
Solution:
Calculate the partial derivatives of u.
⇒
⇒
Sum the second partial derivatives.
⇒
Since the sum is zero, u(x, y) = In(x 2 + y 2 ) 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.
Problem 1: Investigate whether the function u(x,y)=x 3 −3xy 2 +3x 2 −3y 2 +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.
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.
Rule for Harmonic Function is that Laplacian of a harmonic function is zero i.e., ∇ 2 f = 0.
One example of Harmonic Function is f(x, y) = sin (x) cosh (y).
Harmonic functions satisfy Laplace’s equation, whereas non-harmonic functions do not satisfy this equation.
For complex functions, the Cauchy-Riemann equations relate partial derivatives. For f(z) = u(x, y) + iv(x, y), the equations are:
and
