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I am encountering functions of real variable with the following property: $$ f(x) = f(1/x) $$ For example, $$ f(x) = \left(x - \frac{1}{x}\right)\log^{3}{x} \qquad x > 0 $$ Is there a name for this property?

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"Symmetric with respect to reciprocation" is how I would call it – Sasha Oct 4 '11 at 19:25
Invariant under inversion – Andrew Oct 4 '11 at 19:27
This is an observation, not a name. If the domain of $f$ consists of positive numbers, then $g(t) = f(e^t)$ is symmetric about the origin. – Srivatsan Oct 4 '11 at 19:33
@SrivatsanNarayanan: I think you mean $f(e^t)$ is symmetric across the $y$ axis. – Shaun Ault Oct 4 '11 at 20:06
@Shaun Ah, yes. You are correct. I thought of it as a function, rather than as a graph on 2-D. In any case, it might be best to say "$g(t)$ is an even function". – Srivatsan Oct 4 '11 at 20:08

Summary of comments:

  • "Invariant under inversion" is a good name for such functions
  • If the domain consists of positive numbers, one can say instead "$t\mapsto f(e^t)$ is even".
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