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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.

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    $\begingroup$ I think asking which functions are most useful is kinda like asking which numbers are most useful; it probably depends on context and opinion. $\endgroup$ – Jam Aug 12 '14 at 8:57
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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.

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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-

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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

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