Is there anything "special" about elementary functions? I just found an article on Liouville's integrability criterion, which gave me a thought.
What makes functions like $\mathrm{Si}(x)$, $\mathrm{Ei}(x)$, $\mathrm{erfc}(x)$, etc. inherently different from $\sin{x}$, $\log{x}$, etc. ? 
Related question: What is the exact definition for the elementary field, and what is its significance to this question?

The elementary functions are a special kind of closed-form expressions. If $f$ is an elementary function, the following statements are equivalent:


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*$f$ is generated from its only argument variable in a finite number of steps by performing only arithmetic operations, power functions with integer exponents, root functions, exponential functions, logarithm functions, trigonometric functions, inverse trigonometric functions, hyperbolic functions and/or inverse hyperbolic functions.

*$f$ is generated from its only argument variable in a finite number of steps by performing only arithmetic operations, exponentials and/or logarithms.

*$f$ is generated from its only argument variable in a finite number of steps by performing only explicit algebraic functions, exponentials and/or logarithms.
