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MuPAD^{®} provides a variety of tools for handling built-in
mathematical functions such as `sin`

, `cos`

, `exp`

,
and so on. These tools implement the mathematical properties of these
functions. Typical examples are the `float`

conversion routine, the `diff`

differentiation
function, or the `expand`

function,
which you use to manipulate expressions:

float(sin(1)); diff(sin(x), x, x, x); expand(sin(x + 1))

You can say that the mathematical knowledge about the built-in
functions is distributed over several system functions: `float`

knows how to compute
numerical approximations of the sine function, `diff`

knows the derivative of the sine
function, and `expand`

knows
the addition theorems of the trigonometric functions.

When you implement your own function, you can integrate it into the MuPAD system, making other functions aware how to work with this new function correctly. If the new function consists only of built-in MuPAD functions, then you do not need to take extra steps. All MuPAD functions interact correctly with the new function:

f := x -> (x*sin(x)): diff(f(x), x)

However, if you implement a function that is not composed of
the standard MuPAD objects (for example, a new special function),
you must distribute the knowledge about the mathematical meaning of
the new function to standard MuPAD functions, such as `diff`

, `expand`

, `float`

, and so on. This
extra task is necessary for integrating the new function with the
rest of the system. For example, you might want to differentiate an
expression that contains both the new function and some built-in functions,
and such differentiation is only possible via the MuPAD differentiation
routine. Therefore, this routine must know how to handle the new symbol.

MuPAD uses *function environments* (domain
type `DOM_FUNC_ENV`

)
to integrate functions into the system. A function environment stores
special function attributes (`slots`

) in an internal table. Whenever
an overloadable system function, such as `diff`

, `expand`

, or `float`

, encounters an object of type `DOM_FUNC_ENV`

,
it searches the function environment for a corresponding slot. If
a system function finds the appropriate slot, it calls that slot and
returns the value produced by the slot. All built-in MuPAD functions
are implemented as function environments:

domtype(sin), domtype(exp)

You can call a function environment as you would call any MuPAD function or procedure:

sin(1.7), exp(1.0)

Suppose you implement the complete elliptic integral functions
of the first and second kind, `K(z)`

and `E(z)`

.
These functions appear in different contexts, such as calculating
the perimeter of an ellipsis, the gravitational or electrostatic potential
of a uniform ring, and the probability that a random walk in three
dimensions ever goes through the origin. The elliptic integrals have
the following special values:

, *E*(1)
= 1,
,
.

MuPAD provides the built-in functions `ellipticE`

and `ellipticK`

for
computing these elliptic integrals. However, you can implement your
own functions for the same task. For example, write the procedures `ellipE`

and `ellipK`

.
These procedures define the values of the elliptic integrals for special
values of `x`

. For all other argument values, the
values of elliptic integrals are unknown, and the procedures return
the symbolic expressions `ellipE(x)`

and `ellipK(x)`

.
Use `procname`

to
return symbolic expressions:

ellipE := proc(x) begin if x = 0 then PI/2 elif x = 1 then 1 else procname(x) end_if end_proc:

ellipK := proc(x) begin if x = 0 then PI/2 elif x = 1/2 then 8*PI^(3/2)/gamma(-1/4)^2 elif x = -1 then gamma(1/4)^2/4/sqrt(2*PI) else procname(x) end_if end_proc:

`ellipE`

and `ellipK`

return
special values for particular arguments. For all other arguments,
they return symbolic expressions:

ellipE(0), ellipE(1/2), ellipK(12/17), ellipK(x^2 + 1)

The first derivatives of these elliptic integrals are as follows:

, .

The standard MuPAD differentiation function `diff`

does not know about
these rules. Therefore, trying to differentiate `ellipE`

and `ellipK`

simply
returns the symbolic notations of the derivatives:

diff(ellipE(x), x), diff(ellipK(x), x)

To make `diff`

work
with the new functions, create function environments from the procedures `ellipE`

and `ellipK`

.
In addition, function environments let you control the appearance
of the symbolic function calls in outputs.

A function environment consists of three operands.

The first operand is a procedure that computes the return value of a function call.

The second operand is a procedure for printing a symbolic function call on the screen.

The third operand is a table that specifies how the system functions handle symbolic function calls.

To create function environments, use `funcenv`

. For example, create function
environments `ellipE`

and `ellipK`

.
Use the second argument to specify that symbolic calls to `ellipE`

and `ellipK`

must
appear as `E`

and `K`

outputs:

output_E := f -> hold(E)(op(f)): ellipE := funcenv(ellipE, output_E):

output_K := f -> hold(K)(op(f)): ellipK := funcenv(ellipK, output_K):

Although `ellipE`

and `ellipK`

are
now function environments, you can call them as you would call any
other MuPAD function:

ellipE(0), ellipE(1/2), ellipK(12/17), ellipK(x^2+1)

The third argument `funcenv`

is
a table of function attributes. It tells the system functions (such
as `float`

, `diff`

, `expand`

, and so on) how
to handle symbolic calls of the form `ellipE(x)`

and `ellipK(x)`

.
You can update this table specifying the rules for the new function.
For example, specify the new differentiation rules by assigning the
appropriate procedures to the `diff`

slot of the
function environments:

ellipE::diff := proc(f,x) local z; begin z := op(f); (ellipE(z) - ellipK(z))/(2*z) * diff(z, x) end_proc:

ellipK::diff := proc(f,x) local z; begin z := op(f); (ellipE(z) - (1-z)*ellipK(z))/ (2*(1-z)*z) * diff(z, x) end_proc:

Now, whenever `f = ellipE(z)`

, and `z`

depends
on `x`

, the call `diff(f, x)`

uses
the procedure assigned to `ellipE::diff`

:

diff(ellipE(z), z); diff(ellipE(y(x)), x); diff(ellipE(x*sin(x)), x)

The new differentiation routine also finds higher-order derivatives:

diff(ellipE(x), x, x)

Since the `taylor`

function
internally calls `diff`

,
the new differentiation routine also lets you compute Taylor expansions
of the elliptic integrals:

taylor(ellipK(x), x = 0, 6)

If a derivative of a function contains the function itself,
the integration routine has a good chance of finding symbolic integrals
after you implement the `diff`

attributes.
For example, `int`

now
computes the following integrals:

int(ellipE(x), x)

int(ellipK(x), x)

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