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Does anyone know of any even special functions that grow very fast, faster than $\cosh(x)$? (Not the exponential)

(Further info):

$$\sqrt{\ln\left(\cosh\left(m\theta\right)\left(a+b\theta^{2}\right)\right)}$$

is growing too slowly for me. What I'm trying to do is approximate a lower semi-circle using a function of this root log form. The problem is, root ln grows very slowly; in polars this means that if the first and second derivative of the lower semicircle (in polars) and this $\sqrt{\ln\left(f(\theta)\right)}$ function match, the two curves touch 'for a moment', the length of this moment determined, to some extent, by how fast $\cosh(x)$ grows.

Any other candidates that might speed things up, if you understand?

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  • $\begingroup$ preferably one that can be integrated without extreme difficulty $\endgroup$ Commented Mar 15, 2014 at 9:22
  • $\begingroup$ Would $e^x$ be ridiculous ? $\endgroup$ Commented Mar 15, 2014 at 9:23
  • $\begingroup$ ones that aren't even would be fine $\endgroup$ Commented Mar 15, 2014 at 9:24
  • $\begingroup$ not ridiculous, just I need something faster than that (essentially that grows faster than cosh) $\endgroup$ Commented Mar 15, 2014 at 9:25
  • $\begingroup$ Could you eleborate your question ? What kind of property are you looking for ? $\endgroup$ Commented Mar 15, 2014 at 9:26

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You can always try $e^{f(t)}$, for an $f$ of your choice. The exponential will take care of the logarithm, and you can design $f$ to handle the root in whichever way you want.

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