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Simplification of expressions
*INTRO Simplification of expression is a big and nontrivial subject. Simplification implies that there is a preferred form. In practice the preferred form depends on the calculation at hand. This chapter describes the functions offered that allow simplification of expressions.
*CMD Simplify  try to simplify an expression
*STD
*CALL
Simplify(expr)
*PARMS
{expr}  expression to simplify
*DESC
This function tries to simplify the expression {expr} as much
as possible. It does this by grouping powers within terms, and then
grouping similar terms.
*E.G.
In> a*b*a^2/ba^3
Out> (b*a^3)/ba^3;
In> Simplify(a*b*a^2/ba^3)
Out> 0;
*SEE TrigSimpCombine, RadSimp
*CMD RadSimp  simplify expression with nested radicals
*STD
*CALL
RadSimp(expr)
*PARMS
{expr}  an expression containing nested radicals
*DESC
This function tries to write the expression "expr" as a sum of roots
of integers: $Sqrt(e1) + Sqrt(e2) + ...$, where $e1$, $e2$ and
so on are natural numbers. The expression "expr" may not contain
free variables.
It does this by trying all possible combinations for $e1$, $e2$, ...
Every possibility is numerically evaluated using {N} and compared with the numerical evaluation of
"expr". If the approximations are equal (up to a certain margin),
this possibility is returned. Otherwise, the expression is returned
unevaluated.
Note that due to the use of numerical approximations, there is a small
chance that the expression returned by {RadSimp} is
close but not equal to {expr}. The last example underneath
illustrates this problem. Furthermore, if the numerical value of
{expr} is large, the number of possibilities becomes exorbitantly
big so the evaluation may take very long.
*E.G.
In> RadSimp(Sqrt(9+4*Sqrt(2)))
Out> Sqrt(8)+1;
In> RadSimp(Sqrt(5+2*Sqrt(6)) \
+Sqrt(52*Sqrt(6)))
Out> Sqrt(12);
In> RadSimp(Sqrt(14+3*Sqrt(3+2
*Sqrt(512*Sqrt(32*Sqrt(2))))))
Out> Sqrt(2)+3;
But this command may yield incorrect results:
In> RadSimp(Sqrt(1+10^(6)))
Out> 1;
*SEE Simplify, N
*CMD FactorialSimplify  Simplify hypergeometric expressions containing factorials
*STD
*CALL
FactorialSimplify(expression)
*PARMS
{expression}  expression to simplify
*DESC
{FactorialSimplify} takes an expression that may contain factorials,
and tries to simplify it. An expression like $ (n+1)! / n! $ would
simplify to $(n+1)$.
The following steps are taken to simplify:
* 1. binomials are expanded into factorials
* 2. the expression is flattened as much as possible, to reduce it to a sum of simple rational terms
* 3. expressions like $ p^n/p^m $ are reduced to $p^(nm)$ if $nm$ is an integer
* 4. expressions like $ n! / m! $ are simplified if $nm$ is an integer
The function {Simplify} is used to determine if the relevant expressions $nm$
are integers.
*EG
In> FactorialSimplify( (nk+1)! / (nk)! )
Out> n+1k
In> FactorialSimplify(n! / Bin(n,k))
Out> k! *(nk)!
In> FactorialSimplify(2^(n+1)/2^n)
Out> 2
*SEE Simplify, !, Bin
*CMD LnExpand  expand a logarithmic expression using standard logarithm rules
*STD
*CALL
LnExpand(expr)
*PARMS
{expr}  the logarithm of an expression
*DESC
{LnExpand} takes an expression of the form $Ln(expr)$, and applies logarithm
rules to expand this into multiple {Ln} expressions where possible. An
expression like $Ln(a*b^n)$ would be expanded to $Ln(a)+n*Ln(b)$.
If the logarithm of an integer is discovered, it is factorised using {Factors}
and expanded as though {LnExpand} had been given the factorised form. So
$Ln(18)$ goes to $Ln(x)+2*Ln(3)$.
*EG
In> LnExpand(Ln(a*b^n))
Out> Ln(a)+Ln(b)*n
In> LnExpand(Ln(a^m/b^n))
Out> Ln(a)*mLn(b)*n
In> LnExpand(Ln(60))
Out> 2*Ln(2)+Ln(3)+Ln(5)
In> LnExpand(Ln(60/25))
Out> 2*Ln(2)+Ln(3)Ln(5)
*SEE Ln, LnCombine, Factors
*CMD LnCombine  combine logarithmic expressions using standard logarithm rules
*STD
*CALL
LnCombine(expr)
*PARMS
{expr}  an expression possibly containing multiple {Ln} terms to be combined
*DESC
{LnCombine} finds {Ln} terms in the expression it is given, and combines them
using logarithm rules. It is intended to be the exact converse of {LnExpand}.
*EG
In> LnCombine(Ln(a)+Ln(b)*n)
Out> Ln(a*b^n)
In> LnCombine(2*Ln(2)+Ln(3)Ln(5))
Out> Ln(12/5)
*SEE Ln, LnExpand
*CMD TrigSimpCombine  combine products of trigonometric functions
*STD
*CALL
TrigSimpCombine(expr)
*PARMS
{expr}  expression to simplify
*DESC
This function applies the product rules of trigonometry, e.g.
$Cos(u)*Sin(v) = (1/2)*(Sin(vu) + Sin(v+u))$. As a
result, all products of the trigonometric functions {Cos} and {Sin} disappear. The function also tries to simplify the resulting expression as much as
possible by combining all similar terms.
This function is used in for instance {Integrate},
to bring down the expression into a simpler form that hopefully can be
integrated easily.
*E.G.
In> PrettyPrinter'Set("PrettyForm");
True
In> TrigSimpCombine(Cos(a)^2+Sin(a)^2)
1
In> TrigSimpCombine(Cos(a)^2Sin(a)^2)
Cos( 2 * a )
Out>
In> TrigSimpCombine(Cos(a)^2*Sin(b))
Sin( b ) Sin( 2 * a + b )
 + 
2 4
Sin( 2 * a  b )
 
4
*SEE Simplify, Integrate, Expand, Sin, Cos, Tan
