What functions are most useful after the ones learned in high school? I have learnt how to use trig functions, hyperbolic trig functions, exponentials and logs and simple things like polynomials, ellipses, hyperbolas and rational functions but lately when doing calculus I have found that many problems in differential equations and integration cannot be done using the functions I know. I would like to learn a few more functions but I don't know which ones are the most useful and which are at my level. I was thinking of things like the gamma function or Bessel functions for instance. 
Basically, what are the most useful functions I didn't list in the first sentence. It would be best if they have applications in physics.
 A: The error function, for instance, describing the normal or Gaussian distribution, which is used in probability and statistics, would be one such obvious example. Given the fact that Euler's formula, $e^{it}=\cos t+i\sin t$, yields, for $t=x^2$, $e^{ix^2}=\cos x^2+i\sin x^2$, it is only fitting to mention here the Fresnel integrals as well. Other useful special functions are the exponential and trigonometric integrals. Also, I assume, given your mention of the $\Gamma$ function, that you are probably familiar with the beta function as well. Hypergeometric series also some here to mind. Hope this helps.
A: Some functions appear as especially usefull in one or another particular domain of mathematics or physic. As they are often encountered, it was convenient to give them a name in order to be recognized in the books. The most important of them are referenced as so called "Special functions" :
http://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales- 
A: The standard part function. This allows one to correlate an Archimedean continuum (without infinitesimals) and a Bernoullian continuum (with infinitesimals), and provides an intuitive foundation for the study of the calculus. See http://arxiv.org/abs/1407.0233
