3.4.1 Special Functions

Two of the most widespread ambiguities in the field of special functions, although they are not unique to that field, are the roles of parentheses and superscripts. A parenthesis may indicate either grouping or function application. Superscripts may indicate a power, a function parameter or a derivative. Both of these are partly resolved by declaring all mathematical functions (as L^ATEX macros) and incorporating the various sub/superscript parameters into the macro definition. Thus, any remaining superscript can only be a power or derivative. A parenthesis following a function macro probably contains the arguments (and since we know which special function is involved, we know how many arguments to expect), otherwise the parenthesis is for grouping.

This distinction between the function `parameters' (e.g. the sub- and super-scripts) and `arguments' (e.g. the parenthesized, or fenced, arguments) is consistent with L^ATEX conventions. While the macro \sin `names' the sine function (without arguments), one typically `names' the Bessel function by writing $J_{\nu}$ (Strictly speaking, the function $J$ of two arguments, $\nu$ and $z$, is curried to a function of $z$ alone).

Consequently, the L^ATEX macro we have defined for the Bessel function takes the parameters as arguments. Thus, one refers to the Bessel function itself by

$$\verb\vert\BesselJ{\nu}\vert \rightarrow J_{\nu}$$

but the application of the function, say in an expression, would be

$$\verb\vert\BesselJ{\nu}(z)\vert \rightarrow J_{\nu}(z)$$

Since the formatting of sub- and super- and pre-sub-scripts, tends to yield somewhat messy markup, eliminating that markup has the benefit of easing both the author and parser's tasks, as well as standardizing the presentation of these elements.

And of course a non-trivial benefit is that we know which function $J$ is being referred to, and indeed that it is a function at all.