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Does this expression:

$$f(a, b) = \frac{ab}{a^2 + b^2} \equiv \frac{1}{\frac{a}{b} + \frac{b}{a}}$$

have a name?

Comments

I am unaware of a standard name for this expression, but I would refer to it as something like "pseudo-ratio" or "symmetric-ratio" because it shares some of the properties of a ratio, e.g.:

$$f(\lambda a, \lambda b) = f(a, b)$$

and equals to zero if one of the argument is 0 but the other one is not, but, unlike the ratio, it is symmetric, e.g.:

$$f(a, b) = f(b, a)$$

Obviously, there is plenty of functions with such properties.

norok2
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  • This is not of research level, but let me answer. Your expression has no name, but you can give it a name (e.g. norok2's function). –  Oct 14 '17 at 17:59
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    Regarding the "ratio-like properties", the keyword you want is that $f$ is a "homogeneous function of degree $0$". –  Oct 14 '17 at 18:28
  • It's the harmonic mean of $\frac ab$ and $\frac ba$. – Simply Beautiful Art Oct 14 '17 at 18:31
  • @SimplyBeautifulArt Could be named as half the harmonic mean of the ratio of a and b and the ratio of b and a, but also the half of the inverse of the arithmetic mean of the ratio of a and b and the ratio of b and a, but that is not a simple name, its a little more than a description of the operations involved. – norok2 Oct 14 '17 at 19:49
  • @SimplyBeautifulArt Harmonic mean has $2$ in the numerator. – Wojowu Oct 14 '17 at 20:08

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