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Most mathematical expressions can be represented in different, but mathematically equivalent forms. Some of these forms might look simpler, for example, they can be visibly shorter. However, you might prefer other forms of the same expression for numeric computations. There is no general rule as to which form of an expression is the simplest. When solving a particular problem, you can choose the simplest form of an expression for this problem.

Besides the general simplification functions `simplify`

and `Simplify`

, MuPAD^{®} provides
a set of functions for transforming mathematical expressions to particular
forms. The following table helps you choose the function for transforming
your expression to the appropriate form. To see a short description
of a function and a set of examples, click the link in the left column.
To see the detailed help page for a function, click the function name
in the right column.

If a mathematical expression contains terms with the same powers
of a specified variable or expression, the `collect`

function reorganizes the expression
grouping such terms. When calling `collect`

, specify the variables or expressions
that the function must consider as unknowns. The `collect`

function regards the original
expression as a polynomial in the specified unknowns, and groups the
coefficients with equal powers:

collect(x*y^4 + x*z + 2*x^3 + x^2*y*z + 3*x^3*y^4*z^2 + y*z^2 + 5*x*y*z, x)

`collect`

can
consider an expression as the specified unknown. For example, group
the terms of the following trigonometric expression with the equal
powers of `sin(x)`

and `cos(x)`

:

f := cos(x)^4*sin(x) + cos(x)^2*sin(x)^3 + cos(x)^2*sin(x)^2 - sin(x)^4: collect(f, sin(x))

collect(f, cos(x))

The `collect`

function
also can accept several unknowns for collecting terms. If you have
several unknowns, pass them to `collect`

as a list:

collect(a^2*sin(x) - cos(x)^2*sin(x)^3 + cos(x)^2 + a - a^2*sin(x)^4, [a, cos(x)])

MuPAD also provides a function for combining subexpressions
of an original expression. The `combine`

function uses mathematical identities
for the functions you indicate. For example, combine the trigonometric
expression:

combine(2*sin(x)*cos(x), sincos)

If you do not specify a target function, `combine`

uses the identities for powers
wherever these identities are valid:

*a*^{b}*a*^{c}=*a*^{b + c}*a*^{c}*b*^{c}= (*a**b*)^{c}, if`c`

is an integer(

*a*^{b})^{c}=*a*^{bc}, if`c`

is an integer

For example, by default the function combines the following square roots:

combine(sqrt(2)*sqrt(x))

The function does not combine these square roots because the identity is not valid for negative values of variables:

combine(sqrt(x)*sqrt(y))

To combine these square roots, use the `IgnoreAnalyticConstraints`

option:

combine(sqrt(x)*sqrt(y), IgnoreAnalyticConstraints)

As target functions, `combine`

accepts `arctan`

, `exp`

, `gamma`

, `ln`

, `sincos`

,
and other functions. For the complete list of target functions, see
the `combine`

help
page.

For elementary expressions, the `expand`

function transforms the original
expression by multiplying sums of products:

expand((x + 1)*(x + 2)*(x + 3))

`expand`

also
uses mathematical identities between the functions:

expand(sin(5*x))

`expand`

works
recursively for all subexpressions:

expand((sin(3*x) + 1)*(cos(2*x) - 1))

To prevent the expansion of particular subexpressions, pass
these subexpressions to `expand`

as
arguments:

expand((sin(3*x) + 1)*(cos(2*x) - 1), sin(3*x))

To prevent the expansion of all trigonometric subexpressions
in this example, use the option `ArithmeticOnly`

:

expand((sin(3*x) + 1)*(cos(2*x) - 1), ArithmeticOnly)

To present an expression as a product of sums, try the `factor`

function. The
factored form of the following polynomial is visibly shorter than
the original one. The factored form also shows that this polynomial
has only one root `x = -5`

:

factor(x^10 + 50*x^9 + 1125*x^8 + 15000*x^7 + 131250*x^6 + 787500*x^5 + 3281250*x^4 + 9375000*x^3 + 17578125*x^2 + 19531250*x + 9765625)

For sums of rational expressions, `factor`

first computes a common denominator,
and then factors both the numerator and denominator:

f := (x^3 + 3*y^2)/(x^2 - y^2) + 3: f = factor(f)

The function also can factor expressions other than polynomials and rational functions. Internally, MuPAD converts such expressions into polynomials or rational function by substituting subexpressions with identifiers. After factoring the expression with temporary identifiers, MuPAD restores the original subexpressions:

factor((ln(x)^2 - 1)/(cos(x)^2 - sin(x)^2))

By default, `factor`

searches
for polynomial factors with rational numbers. The function does not
factor the expression into a product containing :

factor(x^2 - 2)

To add the constant to
the numbers used in factoring, use the option `Adjoin`

:

factor(x^2 - 2, Adjoin = [sqrt(2)])

The `normal`

function
represents the original rational expression as a single rational term
with expanded numerator and denominator. The greatest common divisor
of the numerator and denominator of the returned expression is 1:

f := (x^3 + 3*y^2)/(x^2 - y^2) + 3: f = normal(f)

`normal`

cancels
common factors that appear in numerator and denominator:

f := x^2/(x + y) - y^2/(x + y): f = normal(f)

The `normal`

function
also handles expressions other than polynomials and rational functions.
Internally, MuPAD converts such expressions into polynomials
or rational functions by substituting subexpressions with identifiers.
After normalizing the expression with temporary identifiers, MuPAD restores
the original subexpressions:

f := (exp(2*x) - exp(2*y))/(exp(3*x) - exp(3*y)): f = normal(f)

The `partfrac`

function
returns a rational expression in the form of a sum of a polynomial
and rational terms. In each rational term, the degree of the numerator
is smaller than the degree of the denominator. For some expressions, `partfrac`

returns visibly
simpler forms, for example:

partfrac((x^6 + 15*x^5 + 94*x^4 + 316*x^3 + 599*x^2 + 602*x + 247)/(x^6 + 14*x^5 + 80*x^4 + 238*x^3 + 387*x^2 + 324*x + 108), x)

The denominators in rational terms represent the factored common denominator of the original expression:

factor(x^6 + 14*x^5 + 80*x^4 + 238*x^3 + 387*x^2 + 324*x + 108)

As an alternative to the general simplifiers `simplify`

and `Simplify`

,
use the `radsimp`

function
to simplify arithmetic expressions involving square roots or other
radicals. For example, simplify the following numeric expression:

f := 3*sqrt(7)/(sqrt(7) - 2); radsimp(f)

When working with complex numbers, you might need to separate
the real and imaginary part of a symbolic expression. To extract the
real and imaginary part of a complex number, you use the `Re`

and `Im`

functions.
For symbolic expressions, these functions return:

Re(tan(x)), Im(tan(x))

Use the `rectform`

function
to split a symbolic expression into its real and imaginary parts:

y := rectform(tan(x))

To extract the real and imaginary parts of `y`

,
use the `Re`

and `Im`

functions:

Re(y); Im(y)

To present an expression in terms of a particular function,
use the `rewrite`

command.
The command uses mathematical identities between functions. For example,
rewrite an expression containing trigonometric functions in terms
of a particular trigonometric function:

sin(x) = rewrite(sin(x), tan); cos(x) = rewrite(cos(x), tan); sin(2*x) + cos(3*x)^2 = rewrite(sin(2*x) + cos(3*x)^2, tan)

Use `rewrite`

to
express the trigonometric and hyperbolic functions in terms of the
exponential function:

sin(x) = rewrite(sin(x), exp); cos(x) = rewrite(cos(x), exp); sinh(x) = rewrite(sinh(x), exp); cosh(x) = rewrite(cosh(x), exp)

The command also expresses inverse hyperbolic functions in terms of logarithms:

arcsinh(x) = rewrite(arcsinh(x), ln); arccosh(x) = rewrite(arccosh(x), ln)

As target functions, `rewrite`

accepts:
direct and inverse trigonometric functions, direct and inverse hyperbolic
functions, `diff`

, `D`

, `erf`

, `exp`

, `fact`

, `gamma`

, `harmonic`

, `piecewise`

,
and more. See the `rewrite`

help
page for the complete list of target functions.