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?
Name that function property
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analysis
functions
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0"Symmetric with respect to reciprocation" is how I would call it – 2011-10-04
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4Invariant under inversion – 2011-10-04
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0This 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. – 2011-10-04
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1@SrivatsanNarayanan: I think you mean $f(e^t)$ is symmetric across the $y$ axis. – 2011-10-04
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1@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". – 2011-10-04