Key Conditions for a Function to Be Harmonic
Harmonic functions that arise in the subject physics are determined by their singularities as well as boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions). On regions without boundaries, adding the real part or the imaginary part of an entire function will produce a harmonic function with the same singularity, so in this case, the harmonic function can not be determined by its singularities; however, we can make the solution unique in physical situations by requiring that the solution approaches 0 as r approaches infinity.
Harmonic functions appear regularly and these functions play a fundamental role in math, physics, as well as in engineering. In this article, we are going to learn the definition, some key properties.
What is Harmonic Function?
Let’s start by defining harmonic functions and looking at some of the properties of harmonic functions.
A function u(x, y) is known as harmonic if it is twice continuously differentiable and satisfies the following partial differential equation:
\[ \nabla ^{2} u = u_{xx} + u_{yy} = 0 \]. (1) Equation 1 is called Laplace’s equation.
So a function is known to be harmonic if it satisfies Laplace’s equation.
Operator \[\nabla ^{2}\] is known as the Laplacian and \[\nabla ^{2} u \] is known as the Laplacian of u.
(Image will be uploaded soon)
What is Conjugate Harmonic Function?
If f(z) = u+iv is known to be an analytic function of z, then v is known as a conjugate harmonic function of u, and u in its turn is termed as a conjugate harmonic function of v.
Or u and v are known as conjugate harmonic functions.
What are Spherical Harmonic Functions?
Spherical harmonic functions generally arise when the spherical coordinate system is used. (In this system, a point in space is located by 3 coordinates, out of which one representing the distance from the origin and two others representing the angles of elevation and azimuth, as in astronomy.) The spherical harmonic functions are commonly used to describe three-dimensional fields, such as they are used to describe magnetic, gravitational, as well as electrical fields.
Why are Harmonic Functions Called Harmonic?
"Harmonic" known to be the descriptor in the name harmonic function, generally originates from a point on a taut string that is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines as well as in terms of cosines, functions which are thus referred to as harmonics.
What are Positive Harmonic Functions?
The Poisson integral formula allows obtaining useful inequalities for positive harmonic functions. Note: It is important to keep in mind that if a non-negative harmonic function attains a minimum value of 0 on a domain, it is 0 throughout the domain. So the class of non-negative harmonic functions on a domain basically consists of all positive functions as well as a zero function.
Condition for Harmonic Function
What Makes a Function Harmonic?
A real-valued function of a single variable is harmonic precisely when it has the form f(x)=ax+b for some numbers a,b, so its graph is a straight line. You can “see” that a graph is a straight line if you have a vision of infinite precision and infinite breadth (you need to make sure the graph doesn’t take a nosedive at x= \[10^{7000}\]), but assuming that straight lines are visibly straight lines then you can do that.
The graph of a function f: \[ R^{2}\rightarrow {R} \] is still something we can visualize – as a surface, or using contour lines – but the harmonic nature of the graph is not something you can “see”. You need the sum of the two-second derivatives to be equal to 0 everywhere, and that doesn’t dictate any easily recognizable shape.
Linear functions or affine functions are still harmonic, and you can recognize their surface graphs as planes, but many functions that aren’t affine are also harmonic. For example, f(x,y)= excos(y) is harmonic, but excos(1.1y) is not.
Can you Visually Tell Them Apart?
You can sometimes recognize that a function is not harmonic by noticing that it has local maxima or local minima. Harmonic functions can’t have such extremal values.
The value of a harmonic function at any point is the average of its values on a sphere (in two variables, for example, a circle) centered at that point.
Harmonic functions of more than 2 variables pose an even more serious visualization challenge since their graphs no longer fit in three dimensions.
Problems to be Solved
Question 1) Verify u(x,y) =\[x^{3} - 3xy^{2} - 5y\] is harmonic in the entire complex plane.
Solution) \[\frac{\delta u}{\delta x}\] =\[3x^{2} - 3y^{2}\], \[\frac{\delta^{2} u}{\delta x^{2}}\]= 6x, \[\frac{\delta u}{\delta y}\]= -6xy-5, \[\frac{\delta^{2} u}{\delta y^{2}}\] = -6x
Therefore,\[ \frac{\delta^{2} u}{\delta x^{2}} + \frac{\delta^{2} u}{\delta y^{2}} \]= 6x-6x = 0
FAQs on Harmonic Function Explained: Concepts & Applications
1. What defines a function as 'harmonic' in mathematics?
In mathematics, a function is defined as harmonic if it is a twice continuously differentiable function that satisfies Laplace's equation. For a function of two variables, u(x, y), this means the sum of its second-order partial derivatives with respect to each variable is zero. The equation is represented as: ∇²u = ∂²u/∂x² + ∂²u/∂y² = 0.
2. How can you test if a given function of two variables is harmonic?
To determine if a function f(x, y) is harmonic, you must check if it satisfies Laplace's equation. The process is as follows:
- First, calculate the second partial derivative of the function with respect to x (fxx).
- Next, calculate the second partial derivative with respect to y (fyy).
- Finally, add these two derivatives together. If the result is fxx + fyy = 0, the function is harmonic. Otherwise, it is not.
3. What are some common examples of harmonic functions?
There are several types of harmonic functions that appear in mathematics and physics. Some key examples include:
- Linear functions: Any function of the form f(x, y) = ax + by is harmonic.
- Polynomials: Functions like f(x, y) = x² - y² and f(x, y) = 2xy are harmonic.
- Exponential and Trigonometric combinations: A function such as f(x, y) = eˣ cos(y) is a classic example.
- Logarithmic functions: The function f(x, y) = ln(x² + y²) is harmonic on any domain not containing the origin.
4. Why are harmonic functions important in fields like physics and engineering?
Harmonic functions are critically important because they describe steady-state conditions and potential fields where no sources or sinks are present. For instance, they are used to model:
- Electrostatic potential in a region free of electric charges.
- Gravitational potential in empty space.
- Steady-state temperature distribution in an object when the heat flow has stabilised.
5. What is the relationship between an analytic function and a harmonic function?
The connection between analytic and harmonic functions is fundamental in complex analysis. If a function f(z) = u(x, y) + iv(x, y) is analytic in a domain, then both its real part, u(x, y), and its imaginary part, v(x, y), are harmonic functions in that domain. Furthermore, u and v are known as harmonic conjugates. This provides a powerful method for constructing harmonic functions from analytic ones.
6. Is a single-variable function like sin(x) harmonic?
No, a single-variable function like f(x) = sin(x) is not considered harmonic in the context of Laplace's equation. The concept of a harmonic function typically applies to functions of two or more variables. For f(x) = sin(x), the 1D Laplace equation would be f''(x) = 0. Since the second derivative is -sin(x), this is not zero for all x. The term 'harmonic motion' describes functions like sine and cosine, but this is a different concept from being a harmonic function that solves ∇²f = 0.
7. What does the Maximum Principle for harmonic functions state?
The Maximum Principle is a key property of harmonic functions. It states that a non-constant harmonic function defined on a connected, bounded open set cannot attain its maximum or minimum value at an interior point of the set. Instead, the maximum and minimum values must occur exclusively on the boundary of the set. This principle has direct physical applications, such as proving that in a steady-state heat distribution, the hottest and coldest points of a body must lie on its surface.
