Harmonic Mean
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
29-6-2019
2149
Harmonic Mean
The harmonic mean 
of 
numbers 
(where 
, ..., 
) is the number 
defined by
 |
(1)
|
The harmonic mean of a list of numbers may be computed in the Wolfram Language using HarmonicMean[list].
The special cases of 
and 
are therefore given by
and so on.
The harmonic means of the integers from 1 to 
for 
, 2, ... are 1, 4/3, 18/11, 48/25, 300/137, 120/49, 980/363, ... (OEIS A102928 and A001008).
For 
, the harmonic mean is related to the arithmetic mean 
and geometric mean 
by
 |
(4)
|
(Havil 2003, p. 120).
The harmonic mean is the special case 
of the power mean and is one of the Pythagorean means. In older literature, it is sometimes called the subcontrary mean.
The volume-to-surface area ratio for a cylindrical container with height 
and radius 
and the mean curvature of a general surface are related to the harmonic mean.
Hoehn and Niven (1985) show that
 |
(5)
|
for any positive constant 
.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 119-121, 2003.
Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151-156, 1985.
Kenney, J. F. and Keeping, E. S. "Harmonic Mean." §4.13 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 57-58, 1962.
Sloane, N. J. A. Sequences A001008/M2885 and A102928 in "The On-Line Encyclopedia of Integer Sequences."
Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995.
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