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The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion.The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics.
A positive function f on the unit disk with f(0) = 1 is harmonic if and only if there is a probability measure μ on the unit circle such that = + ().The formula clearly defines a positive harmonic function with f(0) = 1.
By this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1) H 0 = 0, (2) H x = H x−1 + 1/x for all complex numbers x except the non-positive integers, and (3) lim m→+∞ (H m+x − H m) = 0 for all complex values x.
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the fundamental frequency of a periodic signal. The fundamental frequency is also called the 1st harmonic ; the other harmonics are known as higher harmonics .
Formally, the definition can be stated as follows. Let be a subset of the Euclidean space and let : {} be an upper semi-continuous function.Then, is called subharmonic if for any closed ball (,) ¯ of center and radius contained in and every real-valued continuous function on (,) ¯ that is harmonic in (,) and satisfies () for all on the boundary (,) of (,), we have () for all (,).
The Riemann zeta function is defined for real > by the convergent series = = = + + +, which for = would be the harmonic series. It can be extended by analytic continuation to a holomorphic function on all complex numbers except x = 1 {\displaystyle x=1} , where the extended function has a simple pole .
The concept of harmonic function originates in theories about just intonation.It was realized that three perfect major triads, distant from each other by a perfect fifth, produced the seven degrees of the major scale in one of the possible forms of just intonation: for instance, the triads F–A–C, C–E–G and G–B–D (subdominant, tonic, and dominant respectively) produce the seven ...
Weakly harmonic function; Weyl's lemma (Laplace equation) This page was last edited on 22 November 2022, at 22:46 (UTC). Text is available under the Creative ...