what is harmonic function?

Answer 1

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace's equation


everywhere on U. There also exists a seemingly weaker definition that is equivalent. Indeed a function is harmonic if and only if it is weakly harmonic.

A function that satisfies is said to be subharmonic.

Examples of harmonic functions of two variables are:

the real and imaginary part of any holomorphic function
the function
f(x1, x2) = ln(x12 + x22)
defined on R2 \ {0} (e.g. the electric potential due to a line charge, and the gravity potential due to a long cylindrical mass)
the function f(x1, x2) = exp(x1)sin(x2).
Examples of harmonic functions of three variables are:

the electric potential outside electric charges
the gravity potential outside masses.
Examples of harmonic functions of n variables are:

the constant, linear and affine functions on all of Rn (for example, the electric potential between the plates of a capacitor, and the gravity potential of a slab)
the function f(x1,...,xn) = (x12 + ... + xn2)1 −n/2 on Rn \ {0} for n ≥ 2.

Answer 2

The simplest example of a harmonic is the notion of a "harmonic mean" -- a kind of average. Suppose you have two numbers, x and y: if you want to express their average, you would calculate (x+y)/2. THis kind of "average", probably the most common, is called the ARITHMETIC MEAN. In some examples -- including the strength of wind and the loudness of sounds, numbers are related by multiplication: 4 is worth twice 3, and 5 is worth twice 4. This also happens in such things as relative distance of orbits around the sun. In these cases the best kind of average is the GEOMETRIC MEAN. The geometric mean of two numbers a and b is square root(a * b)

If you have two resistors connected in parallel in an electric circuit, one may be measured at 40 ohms and the other at 60 ohms. The "average" of these two would be the value of a single resistor you might substitute for them. The arithmetic mean of 40 and 60 is 50 -- but that is not the correct average to use in this case. Nor is 48.989 (the approximate geometric mean of 40 and 60). The appropriate average to use is the HARMONIC MEAN, calculated as the reciprocal of the arithmetic mean of the reciprocals. In simpler terms this is 2ab/(a + b): the harmonic mean of 40 and 60 is 48. Another way of calculating the harmonic mean of two numbers is to calculate G^2/A where G is the geometric mean and A is the arithmetic mean. It is called "harmonic" because the relationship between the different numbers is like the relationship between musical tones.
A "harmonic series" is one in which the reciprocals of the terms are themselves an arithmetic series. An example is 1/1 + 1/2 + 1/3 + 1/4, because 1 + 2 + 3 + 4 has terms in arithmetic progression.
If you google "harmonic series" you will find some sites that deal with maths and others that deal with music theory -- the maths behind the music is the sme!

Answer 3

Ive thought of harmonic functions as subsets of fourier series... made up of a linear combination of sine and cosine functions.

Answer 4

a harmonic function is a twice continuously differentiable function

Answer 5

I know only harmonic progression.

Answer 6

Detailed information on this is available on the following page :

http://en.wikipedia.org/wiki/Harmonic_function