Parametric Operator - Mathematical Functions

Mathematical Functions

Mathematical functions have one or more arguments that are designated in the definition by variables, while their definition can also contain parameters. The variables are mentioned in the list of arguments that the function takes, but the parameters are not. When parameters are present, the definition actually defines a whole family of functions, one for every valid set of values of the parameters. For instance, one could define a general quadratic function by defining

;

here, the variable x designates the function's argument, but a, b, and c are parameters that determine which quadratic function one is considering. The parameter could be incorporated into the function name to indicate its dependence on the parameter. For instance, one may define the base a of a logarithm by

where a is a parameter that indicates which logarithmic function is being used. It is not an argument of the function, and will, for instance, be a constant when considering the derivative .

In some informal situations it is a matter of convention (or historical accident) whether some or all of the symbols in a function definition are called parameters. However, changing the status of symbols between parameter and variable changes the function as a mathematical object. For instance, the notation for the falling factorial power

,

defines a polynomial function of n (when k is considered a parameter), but is not a polynomial function of k (when n is considered a parameter). Indeed, in the latter case, it is only defined for non-negative integer arguments. More formal presentations of such situations typically start out with a function of several variables (including all those that might sometimes be called "parameters") such as

as the most fundamental object being considered, then defining functions with fewer variables from the main one by means of currying.

Sometimes it's useful to consider all functions with certain parameters as parametric family, i.e. as an indexed family of functions. Examples from probability theory are given further below.