expand
Expand an expression
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expand(f
,options
) expand(f, g1, g2, …
,options
)
expand(f)
expands the arithmetical expression f
.
The most important use of expand
is the application
of the distributivity law to rewrite products of sums as sums of products.
In this respect, expand
is the inverse function
of factor
.
The numerator of a fraction is expanded, and then the fraction
is rewritten as a sum of fractions with simpler numerators; see Example 1. In a certain sense,
this is the inverse functionality of normal
. Use partfrac
for a more
powerful way to rewrite a fraction as a sum of simpler fractions.
expand(f)
applies the following rules when
rewriting powers occurring as subexpressions in f
:
x^{a + b} = x^{a} x^{b}.
If b
is an integer, or x ≥
0 or y ≥
0, then (x y)^{b} = x^{b} y^{b}.
If b
is an integer, then (x^{a})^{b} = x^{a b}.
Except for the third rule, this behavior of expand
is
the inverse functionality of combine
.
See Example 2.
expand
works recursively on the subexpressions
of an expression f
. If f
is
of the container type array
or table
, expand
only
returns f
and does not map on the entries. To expand
all entries of one of the containers, use map
. See Example 3.
If optional arguments g1, g2, ...
are present,
then any subexpression of f
that is equal to one
of these additional arguments is not expanded; see Example 4. See section “Background”
for a description how this works.
Properties of
identifiers are taken into account (see assume
). Identifiers without any properties
are assumed to be complex. See Example 9.
expand
also handles various types of special
mathematical functions. It rewrites a single call of a special function
with a complicated argument as a sum or a product of several calls
of the same function or related functions with simpler arguments.
In this respect, expand
is the inverse function
of combine
.
In particular, expand
implements the functional
equations of the exponential function and
the logarithm, the gamma function and the polygamma function, and the addition theorems
for the trigonometric functions and
the hyperbolic functions. See Example 10.
expand
is sensitive to properties of
identifiers set via assume
.
expand
expands products of sums by multiplying
out:
expand((x + 1)*(y + z)^2)
After expansion of the numerator, a fraction is rewritten as a sum of fractions:
expand((x + 1)^2*y/(y + z)^2)
A power with a sum in the exponent is rewritten as a product of powers:
expand(x^(y + z + 2))
expand
works in a recursive fashion. In the
following example, the power (x + y)^{z +
2} is first expanded into a product
of two powers. Then the power (x + y)^{2} is
expanded into a sum. Finally, the product of the latter sum and the
remaining power (x + y)^{z} is
multiplied out:
expand((x + y)^(z + 2))
Here is another example:
expand(2^((x + y)^2))
expand
maps on the entries of lists, sets,
and matrices:
expand([(a + b)^2, (a  b)^2]); expand({(a + b)^2, (a  b)^2}); expand(matrix([[(a + b)^2, 0],[0, (a  b)^2]]))
expand
does not map on the entries of tables
or arrays:
expand(table((a + b)^2=(c + 1)^2)), expand(array(1..1, [(a + b)^2]))
Use map
in
order to expand all entries of a container:
map(table((a + b)^2=(c + 1)^2), expand), map(array(1..1, [(a + b)^2]), expand)
Note that this call expands only the entries in a table, not the keys. In the (rare) case that you want the keys expanded as well, transform the table to a list or set of equations first:
T := table((a + b)^2=(c + 1)^2): table(expand([op(T)]))
If additional arguments are provided, expand
performs
only a partial expansion. These additional expressions, such as x
+ 1
in the following example, are not expanded:
expand((x + 1)*(y + z))
expand((x + 1)*(y + z), x + 1)
By default, expand
works on all subexpressions
including trigonometric subexpressions:
e := (sin(2*x) + 1)*(1  cos(2*x)): expand(e)
To prevent expansion of subexpressions, use the ArithmeticOnly
option:
expand(e, ArithmeticOnly)
The option does not prevent expansion of powers and roots:
expand((sin(2*x) + 1)^3, ArithmeticOnly)
To keep subexpressions with integer powers unexpanded, use the MaxExponent
option.
The IgnoreAnalyticConstraints
option applies
a set of purely algebraic simplifications including the equality of
sum of logarithms and a logarithm of a product. Using the IgnoreAnalyticConstraints
option,
you get a simpler result, but one that might be incorrect for some
of the values of variables:
expand(ln(a*b*c*d), IgnoreAnalyticConstraints)
Without using this option, you get a mathematically correct result:
expand(ln(a*b*c*d))
If the additional MaxExponent
provided, expand
performs
only a partial expansion. Powers with an integer exponent larger than
the given bound, are not expanded:
expand((a + b)^3, MaxExponent = 2)
If the exponent is smaller or equal the given bound, the power is expanded:
expand((a + b)^2, MaxExponent = 2)
The expand
function can accept several options
simultaneously. Suppose you want to expand the following expression:
e := (sin(2*x) + 1)*(x + 1)^3
expand
without any options works recursively.
The function expands all subexpressions including trigonometric functions
and powers:
expand(e)
The ArithmeticOnly
option prevents the expansion
of the term sin(2x)
. The MaxExponent
option
prevents the expansion of (x +
1)^{3}:
expand(e, ArithmeticOnly); expand(e, MaxExponent = 2)
Combining these options in one call of the expand
function,
you apply both restrictions for the expansion:
expand(e, MaxExponent = 2, ArithmeticOnly)
The following expansions are not valid for all values a
, b
from
the complex plane. Therefore, MuPAD^{®} does not expand these expressions:
expand(ln(a^2)), expand(ln(a*b)), expand((a*b)^n)
The expansions are valid under the assumption that a
is
a positive real number:
assume(a > 0): expand(ln(a^2)), expand(ln(a*b)), expand((a*b)^n)
Clear the assumption for further computations:
unassume(a):
Alternatively, to get the expanded result for the third expression,
assume that n
is an integer:
expand((a*b)^n) assuming n in Z_
Use the IgnoreAnalyticConstraints
option
to expand these expressions without explicitly specified assumptions:
expand(ln(a^2), IgnoreAnalyticConstraints), expand(ln(a*b), IgnoreAnalyticConstraints), expand((a*b)^n, IgnoreAnalyticConstraints)
The addition theorems of trigonometry are implemented by "expand"
slots
of the trigonometric functions sin
and cos
:
expand(sin(a + b)), expand(sin(2*a))
The same is true for the hyperbolic
functions sinh
and cosh
:
expand(cosh(a + b)), expand(cosh(2*a))
The exponential function with
a sum as argument is expanded via exp::expand
:
expand(exp(a + b))
Here are some more expansion examples for the functions sum
, fact
, abs
, coth
, sign
, binomial
, beta
, gamma
, cot
, tan
, exp
and psi
:
sum(f(x) + g(x),x); expand(%)
fact(x + 1); expand(%)
abs(a*b); expand(%)
coth(a + b); expand(%)
coth(a*b); expand(%)
sign(a*b); expand(%)
binomial(n, m); expand(%)
beta(n, m); expand(%)
gamma(x + 1); expand(%)
tan(a + b); expand(%)
cot(a + b); expand(%)
exp(x + y); expand(%)
psi(x + 2); expand(%)
In contrast to previous versions of MuPAD, expand
does
not rewrite tan
in
terms of sin
and cos
:
expand(tan(a))
This example illustrates how to extend the functionality of expand
to
userdefined mathematical functions. As an example, we consider the
sine function. (Of course, the system function sin
already has an "expand"
slot;
see Example 10.)
We first embed our function into a function environment, which
we call Sin
, in order not to overwrite the system
function sin
.
Then we implement the addition theorem sin(x + y)
= sin(x) cos(y)
+ sin(y) cos(x) in
the "expand"
slot of the function environment,
i.e., the slot routine Sin::expand
:
Sin := funcenv(Sin): Sin::expand := proc(u) // compute expand(Sin(u)) local x, y; begin // recursively expand the argument u u := expand(op(u)); if type(u) = "_plus" then // u is a sum x := op(u, 1); // the first term y := u  x; // the remaining terms // apply the addition theorem and // expand the result again expand(Sin(x)*cos(y) + cos(x)*Sin(y)) else Sin(u) end_if end_proc:
Now, if expand
encounters a subexpression
of the form Sin(u)
, it calls Sin::expand(u)
to
expand Sin(u)
. The following command first expands
the argument a*(b+c)
via the recursive call in Sin::expand
,
then applies the addition theorem, and finally expand
itself
expands the product of the result with z
:
expand(z*Sin(a*(b + c)))
The expansion after the application of the addition theorem
in Sin::expand
is necessary to handle the case
when u
is a sum with more than two terms: then y
is
again a sum, and cos(y)
and Sin(y)
are
expanded recursively:
expand(Sin(a + b + c))


Expand arithmetic part of an expression without expanding trigonometric, hyperbolic, logarithmic and special functions. This option does not prevent expansion of powers and roots.
Technically, the option omits overloading the 

With this option
Using the option can give you simpler results for the expressions
for which the default call to 

Option, specified as Do not expand powers with integer exponents larger than If you call 
f
With optional arguments g1, g2, ...
, the
expansion of certain subexpressions of f
can be
prevented. This works as follows: every occurrence of g1,
g2, ...
in f
is replaced by an auxiliary
variable before the expansion, and afterwards the auxiliary variables
are replaced by the original subexpressions.
Users can extend the functionality of expand
to
their own special mathematical functions via overloading.
To this end, embed your function into a function
environment g
and implement the behavior
of expand
for this function in the "expand"
slot
of the function environment.
Whenever expand
encounters a subexpression
of the form g(u,..)
, it issues the call g::expand(g(u,..))
to
the slot routine to expand the subexpression, passing the not yet
expanded arguments g(u,..)
as arguments. The result
of this call is not expanded any further by expand
.
See Example 11 above.
Similarly, an "expand"
slot can be defined
for a userdefined library domainT
. Whenever expand
encounters
a subexpression d
of domain typeT
,
it issues the call T::expand(d)
to the slot routine
to expand d
. The result of this call is not expanded
any further by expand
. If T
has
no "expand"
slot, then d
remains
unchanged.