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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.
For all positive data sets containing at least one pair of nonequal values, the harmonic mean is always the least of the three Pythagorean means, [5] while the arithmetic mean is always the greatest of the three and the geometric mean is always in between. (If all values in a nonempty data set are equal, the three means are always equal.)
Total harmonic distortion, or THD is a common measurement of the level of harmonic distortion present in power systems. THD can be related to either current harmonics or voltage harmonics, and it is defined as the ratio of the RMS value of all harmonics to the RMS value of the fundamental component times 100%; the DC component is neglected.
where V n is the RMS value of the nth harmonic voltage, and V 1 is the RMS value of the fundamental component. In practice, the THD F is commonly used in audio distortion specifications (percentage THD); however, THD is a non-standardized specification, and the results between manufacturers are not easily comparable. Since individual harmonic ...
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
For any Borel subset E of ∂D, the harmonic measure ω(x, D)(E) is equal to the value at x of the solution to the Dirichlet problem with boundary data equal to the indicator function of E. For fixed D and E ⊆ ∂D, ω(x, D)(E) is a harmonic function of x ∈ D and (,) ();
The random harmonic series is =, where the values are independent and identically distributed random variables that take the two values + and with equal probability . It converges with probability 1 , as can be seen by using the Kolmogorov three-series theorem or of the closely related Kolmogorov maximal inequality .
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.