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In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.. It is the most appropriate average for ratios and rates such as speeds, [1] [2] and is normally only used for positive arguments.
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.
A geometric construction of the quadratic mean and the Pythagorean means (of two numbers a and b). Harmonic mean denoted by H, geometric by G, arithmetic by A and quadratic mean (also known as root mean square) denoted by Q. Comparison of the arithmetic, geometric and harmonic means of a pair of numbers.
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) [1] are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means ( arithmetic , geometric , and harmonic means ).
NumPy (pronounced / ˈ n ʌ m p aɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3]
The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on the real axis . The only one on the positive real axis is the unique minimum of the real-valued gamma function on R + at x 0 = 1.461 632 144 968 362 341 26 ... .
Each function above will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The inversion of each function will yield another harmonic function which has singularities which are the images of the original singularities in a spherical "mirror". Also, the sum of any two harmonic functions will ...
It is not monotonic − increasing a value of can decrease the value of the contraharmonic mean. For instance C(1, 4) > C(2, 4).. The contraharmonic mean is higher in value than the arithmetic mean and also higher than the root mean square: () where x is a list of values, H is the harmonic mean, G is geometric mean, L is the logarithmic mean, A is the arithmetic mean, R ...