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I am wondering if there is a function $f(x)$ "similar" to the exponential function $\exp(x)$ such that:

$-f(x) \approx f(-x)$

I would also like $f(x)$ to have the following property:

$\frac{{f(a)}}{{f(b)}} = f(a - b)$

Or alternately,

$\frac{{f(a)}}{{f(b)}} \approx f(a - b)$

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    @RahulNarain: That's$a$really interesting little proof. Thanks for clarifying this. I suppose that I am going to have to settle for $f(a)/f(b) \approx f(a-b)$2012-07-28

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You might be interested in the hyperbolic sine "sinh". It is antisymmetric and its asymptotic behaviour for $x\to\infty$ is similar to the one of the exponential function.

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    @NicholasKinar Nice!2012-07-28