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The function

$$\frac{1}{\;\;\dfrac{1}{x} + \dfrac1{y}\;\;}$$

shows up a lot, e.g., in parallel resistance or series conductance. Does it have a name? It is similar to harmonic mean with the difference that the numerator is one rather than the number of terms.

Can it be called harmonic sum? I think it stands to reason that since:

  • you can replace $n$ series resistors with $n$ equal resistors having resistance equal to the mean of those resistors,
  • you can replace $n$ parallel resistors with $n$ equal reistors having resistance equal to the harmonic mean of those resistance, and
  • you can replace $n$ series resistor with 1 resistor having resistance equal to the sum of those resistors, then

— by analogy — you should be able to replace $n$ parallel resistors with 1 resistor having resistance equal to the “harmonic sum” of those resistors.

That is, mean is to harmonic mean as sum is to “harmonic sum”.

Neil G
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  • I remember the equation $\frac{1}{h}=\frac{1}{x}+\frac{1}{y}$ occuring in optics (with other letters), but I do not know what "h" means mathematically. – Peter Dec 12 '16 at 00:03
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    @Peter: The numerator needs to be 2 if we want the harmonic mean of a number and itself to be that number again. – hmakholm left over Monica Dec 12 '16 at 00:05
  • OK , I must have mixed something. The "h" in my equation should have a similar meaning – Peter Dec 12 '16 at 00:05
  • @Blue I know the definition of harmonic mean. I mentioned it in the question! And if there are three terms the numerator would have to be 3, etc. The question is is there a name for this pattern when the numerator does not depend on the number of terms. – Neil G Dec 12 '16 at 00:05
  • @Blue Ah, cool. Feel free to edit the question :) – Neil G Dec 12 '16 at 00:12
  • I don't think the operation has an established name. It is common enough, for instance when calculating the resistance of resistors in parallel, or the total time for two people to finish a set amount of work given the time they would each take it they were alone, that if it had a name, I think we would've heard about it. – Arthur Dec 12 '16 at 00:14
  • The subsection of the Wikipedia article on harmonic mean concerning physics has a discussion comparing the roles of harmonic mean for parallel resistance and arithmetic mean for series resistance. That discussion makes a point in a way that does generalize from two resistors to any fixed number. – hardmath Dec 12 '16 at 00:16
  • @hardmath Thanks for sharing that. I had read it before posting, and I now edited the question to explain my logic. – Neil G Dec 12 '16 at 00:23
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    "half of the harmonic mean"? – ypercubeᵀᴹ Dec 12 '16 at 00:28
  • Maybe call it 'reduced harmonic mean' ? Sounds kinda nice and it is also connected to reduced masses. – Hyperplane Dec 12 '16 at 00:39
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    It is also the answer to the question: If one worker can perform a certain task in x time-units and another can perform the same task in y time-units; then if the two workers are to perform the task in tandem, how many time-units will be required? – Senex Ægypti Parvi Dec 12 '16 at 01:10
  • https://en.wikipedia.org/wiki/Harmonic_mean#In_physics – David Aug 20 '19 at 15:58
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    Have a look at https://math.stackexchange.com/questions/1785715 – Hyperplane Aug 20 '19 at 17:33

1 Answers1

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If there were not already an established meaning for "harmonic sum", your choice of a name for the expression $\frac{1}{\frac{1}{x}+\frac{1}{y}}$ would be suitable.

Unfortunately harmonic sum is already widely used in connection with the harmonic series and its partial sums.

Even so, if you wished to reuse this phrase or introduce a novel one for this type of expression, you are free to do so in mathematical exposition if you provide a definition and make clear that your choice is not to be confused with conventional terminology. For example, if pressed I might be tempted to call these sorts of values unweighted harmonic means. The phrase is not self-explanatory, but it is odd enough to motivate a careful Reader to check my definition and see that it differs from the usual harmonic mean.

hardmath
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