Generating Special Functions. I find it quite intriguing that I can generate a lot of special functions, my questions are as follows:

*

*Is there any limit on the number of special functions?


*What current literature in applied and rigorous Special Functions can you recommend me?


*I looked at the integral $$\int \frac{log x}{Ei(x)}$$ in mathematica, and there doesn't seem to be a lot of information regarding this function, does it have any applications in analysis or mathematical physics?
Thanks in advance.
 A: *

*There is no limit on the number of functions you can explicitly name.  But a function does not get to be called 'special' just if someone can name it!  It has to recur a sufficient number of times, in different applications, to acquire that 'special' moniker.  The addition of new special functions to various compendiums like Abramowitz and Stegun has slowed down dramatically over the years.  Even its electronic companion (easier to update!), the DLMF, has few new additions.  My favourite 'new' special functions are definitely the Heun Functions as well as the Lambert W function.

*[I will come back later and edit this part of my answer, as I don't have the references handy right now].  There is some solid literature that shows that a lot of special functions arise quite naturally as solutions of (initial value problems for) linear ODEs with polynomial coefficients.  Most of the named functions in fact correspond (exactly!) to a particular placement of singularities, each with specific local behaviour.  There are decent explanations for most of the special functions (i.e. they are not 'randomly named' functions at all, but would have inevitably have been discovered).

*Very unlikely.  Just because you can name it doesn't mean that it shows up anywhere.  

