Expressions with Heads That Are Not Symbols
In most cases, you want the head
f of a
Mathematica expression like
to be a single symbol. There are, however, some important applications of heads that are not symbols.
This expression has
as a head. You can use heads like this to represent "indexed functions".
| Out[1]= |  |
You can use any expression as a head. Remember to put in the necessary parentheses.
| Out[2]= |  |
One case where we have already encountered the use of complicated expressions as heads is in working with pure functions in
"Pure Functions". By giving
Function
as the head of an expression, you specify a function of the arguments to be evaluated.
With the head
Function
, the value of the expression is the square of the argument.
| Out[3]= |  |
There are several constructs in
Mathematica which work much like pure functions, but which represent specific kinds of functions, typically numerical ones. In all cases, the basic mechanism involves giving a head which contains complete information about the function you want to use.
Some expressions which have heads that are not symbols.
| Out[4]= |  |
| Out[5]= |  |
You can use the
InterpolatingFunction object as a head to get numerical approximations to values of the function
.
| Out[6]= |  |
Another important use of more complicated expressions as heads is in implementing
functionals and
functional operators in mathematics.
As one example, consider the operation of differentiation. As discussed in
"The Representation of Derivatives", an expression like
represents a
derivative function, obtained from
by applying a functional operator to it. In
Mathematica,
is represented as
Derivative[1][f]: the "functional operator"
Derivative
is applied to
to give another function, represented as
.
This expression has a head which represents the application of the "functional operator"
Derivative
to the "function"
.
Out[7]//FullForm= |
| |  |
You can replace the head
with another head, such as
. This effectively takes
to be a "derivative function" obtained from
.
| Out[8]= |  |
