The [modified Bessel function of the 1st kind][1] $I_0$ is defined by
$$
I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta
$$
and arises, among other places, in the probability density function of a noncentral $\chi^2(2)$ random variable.

The [error function][2] is defined by
$$
\text{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2}dt
$$
and is closely related to the cumulative distribution function for the standard normal distribution.

I believe these are not [elementary functions][3], but
>is either $I_0$ or $\text{erf}$ elementary "**relative to**" the other?

That is, if we were to add one of them to our repertoire of "elementary" functions, would the other one become "elementary"?


  [1]: http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html
  [2]: http://en.wikipedia.org/wiki/Error_function
  [3]: http://en.wikipedia.org/wiki/Elementary_function