Hyperbolic functions: Difference between revisions

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===Inequalities===

<math>\operatorname{cosh}(t)The \leqfollowing e^{t^2inequality /2}</math>is useful in statistics:<ref>{{cite news |last1=Audibert |first1=Jean-Yves |date=2009 |title=Fast learning rates in statistical inference through aggregation |publisher=The Annals of Statistics |page=1627}} [https://projecteuclid.org/download/pdfview_1/euclid.aos/1245332827]</ref>

The following inequality is useful in statistics:

<math display="block">\operatorname{cosh}(t) \leq e^{t^2 /2}.</math>

<math>\operatorname{cosh}(t) \leq e^{t^2 /2}</math> <ref>{{cite news |last1=Audibert |first1=Jean-Yves |date=2009 |title=Fast learning rates in statistical inference through aggregation |publisher=The Annals of Statistics |page=1627}} [https://projecteuclid.org/download/pdfview_1/euclid.aos/1245332827]</ref>

It can be proved by comparing term by term the Taylor series of the two functions term by term.

==Inverse functions as logarithms==