What field of mathematics does one first get introduced to non-elementary functions? Knowing not much else other than basic linear algebra, single-variable and multivariable calculus, I would like to expand my mathematical knowledge .
I've always found non-elementary functions, such as the gamma function and particularly the Riemann zeta function, interesting. However I am not sure where to start learning about them. I want to avoid jumping right in due to a gap in my knowledge, so I am here to ask:
In what area of math does one start getting acquainted to functions such as the ones mentioned above? What would be prerequisite to start learning, and what textbooks would be useful (if you happen to know)? 
Thanks, and if my question seems too personal, please let me know and I will reword it accordingly. 
 A: Special functions typically arise in some subfield of analysis. As anon has stated, they are often seen in some context of complex analysis and differential equations, but one sometimes sees them in other contexts as well.
In differential equations one comes across special functions as solutions to certain differential equations. The Bessel functions (https://en.wikipedia.org/wiki/Bessel_function) are among the most important examples. As you can see from the link, they arise in a wealth of different contexts. The Airy function (https://en.wikipedia.org/wiki/Airy_function) is another example of a special function that is primarily defined as a solution to a differential equation.
A vast collection of special functions arise as or are related to the hypergeometric functions (https://en.wikipedia.org/wiki/Hypergeometric_function). They are seen in several contexts, including combinatorics and number theory.
In probability and statistics, the importance of the normal distribution (https://en.wikipedia.org/wiki/Normal_distribution) naturally leads one to consider the error function (https://en.wikipedia.org/wiki/Error_function), among related special functions.
Several of these functions are also seen outside of these contexts. For example, a harmonic analyst comes across Bessel functions when one considers the Fourier transform of a radially symmetric function, and the normal distribution as eigenfunctions of the Fourier transform.
In complex analysis one comes across a wealth of special functions, often in connection with number theory. The Riemann zeta (https://en.wikipedia.org/wiki/Riemann_zeta_function) is a well-known example, and through functional equations one encounters it in conjunction with the gamma function (https://en.wikipedia.org/wiki/Gamma_function). Theta functions (https://en.wikipedia.org/wiki/Theta_function) form a large class of special functions that are seen in a variety of areas.
Non-elementary functions are also studied algebraically in differential algebra (https://en.wikipedia.org/wiki/Differential_algebra). One interesting result in this field is Liouville's theorem, which is useful for computer algebra systems.
Pretty much all of these functions are somehow relevant to theoretical physics, so if you go in that direction you are bound to see at least a few. (See https://en.wikipedia.org/wiki/Zeta_function_regularization for a particularly well-known example.)
If you are interested in special functions, then given the vast number of potentially interesting functions you will have to naturally narrow your focus. Googling for references on "special functions" as a whole will likely lead you to a decent overview of the main functions from physics and engineering. Searching for any of the individual special functions I have listed or linked will also likely lead to a reference on that particular function; however, many of these are likely to be out of your reach at the present given your level of mathematical maturity. For those, the best you can do is to begin an intensive study of analysis and differential equations; if you go far enough in this direction, then you will come across special functions eventually.
There is, however, at least one well-regarded book on the gamma function that is written for readers with only knowledge of basic calculus: The Gamma Function, by Emil Artin. This may be a good start for someone of your preparation.
