Most important non-elementary functions in Math

Question

I am looking for non-elementary function that is

• Well behaved over $\mathbb{R}$ and $\mathbb{C}$ like piecewise smooth, not like Weierstrass function
• Ubiquitous and well studied
• Simple to compute approximate value in a point. W function is better, than a slowly converging series

So far I find elliptic integrals, Gamma function, W function and an integral of a Gaussian function.

What other functions meet requirements?

Answer 1

I would go out on a limb and say that the Bessel Functions are incredibly important non-elementary functions. The Bessel Functions are two classes of functions that solve the differential equation

$$ x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+(x^2-\alpha^2)y=0 $$

for some parameter $\alpha \in \mathbb{C}$. Since it is a second order differential equation we get two solutions: The Bessel Function of the first kind $J_\alpha$, and the Bessel Function of the second kind $Y_\alpha$

This appears all the time in the study of partial differential equations. The first time I personally encountered it in a class was in solving the wave equation for a vibrating drum head. Nice to notice is that $J_\alpha(0)=0$ and $Y_\alpha(0)=-\infty$