New Elementary Function? In the February 2000 issue of FOCUS magazine, a short article suggests that the Lambert W function could be introduced into curriculum as a new elementary function saying: "... a case can be made for according it equal respect with the traditional transcendentals of calculus." As the inverse of $xe^x$, Lambert W is easy to understand, its properties are rather straight-forward, and it has found use in a wide range of applications. 
Are there other functions you think are good candidates to introduce more widely into mathematics curriculum that are interesting, easy to understand, and broadly applicable?
 A: I think one of the candidates may be the error function. It is defined as
$$\operatorname{erf}(x)=\frac{2}{\sqrt\pi}\int_0^xe^{-t^2}\,dt$$
It is easy to understand as an integral of $e^{-x^2}$. The constant $2/\sqrt\pi$ comes from the fact that
$$\int_0^\infty e^{-x^2}\,dx=\frac{\sqrt\pi}{2}$$
and it forces the function to have limit at $\infty$ equal to $1$. Its derivative is elementary, namely
$$\operatorname{erf}'(x)=\frac{2e^{-x^2}}{\sqrt\pi}$$
and its integral may be expressed using elementary function and this function itself,
$$\int\operatorname{erf}(x)\,dx=x\operatorname{erf}(x)+\frac{e^{-x^2}}{\sqrt\pi}+C$$
It is used in probability and calculus.
A: There is a class of differentiated gamma functions (DGFs) that basically can be expressed via derivatives of Gamma function.
They include the Gamma function, Polygamma function, and Hurwitz Zeta function further generalizes the class to fractional orders.
The thing about this class of functions is that they behave like "nearly-elementary".
These are the arguments for this point:

*

*DGFs generalize the Bernoulli polynomials to negative orders, particularly, $\psi^{(1)}(x)=B_{-1}(x)$ if we define Bernoulli polynomials via Hurwitz Zeta. With power functions it is the reciprocal functions that generalize them to negative orders, and considered elementary.

*The antiderivative of the reciprocal function $\frac1x$ is logarithm, an elementary function. The antidifference of the same function is polygamma $\psi(x)$, a DGF.

*$\psi(x)-\psi(-x)$ is elementary, while $\psi(x)+\psi(-x)$ is a DGF. The Taylor series for these functions differ only in parity. Also, $\frac1{\pi}\psi(1-\frac x\pi)-\frac1{\pi}\psi(\frac x\pi)=\cot x$.

*Due to Gauss's digamma theorem, the digamma function of a rational argument may be expressed in terms of Euler's constant and a finite number of elementary functions:$$\psi\left(\frac{r}{m}\right) = -\gamma -\ln(2m) -\frac{\pi}{2}\cot\left(\frac{r\pi}{m}\right) +2\sum_{n=1}^{\left\lfloor \frac{m-1}{2} \right\rfloor} \cos\left(\frac{2\pi nr}{m} \right) \ln\sin\left(\frac{\pi n}{m}\right)$$
Given the above considerations, the DGFs look like "all-but-elementary" or "nearly elementary" to me.
