(directly go to documentation on : Simplify, RadSimp, FactorialSimplify, LnExpand, LnCombine, TrigSimpCombine. )

# 11. Simplification of expressions

Simplification of expression is a big and non-trivial 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.

 Simplify try to simplify an expression RadSimp simplify expression with nested radicals FactorialSimplify Simplify hypergeometric expressions containing factorials LnExpand expand a logarithmic expression using standard logarithm rules LnCombine combine logarithmic expressions using standard logarithm rules TrigSimpCombine combine products of trigonometric functions

### Simplify -- try to simplify an expression

##### Calling format:
 Simplify(expr) 

##### Parameters:
expr -- expression to simplify

##### Description:
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.

##### Examples:
 In> a*b*a^2/b-a^3 Out> (b*a^3)/b-a^3; In> Simplify(a*b*a^2/b-a^3) Out> 0; 

##### Calling format:
 RadSimp(expr) 

##### Parameters:
expr -- an expression containing nested radicals

##### Description:
This function tries to write the expression "expr" as a sum of roots of integers: $\sqrt{e_{1}} + \sqrt{e_{2}} + \mathrm{ ... }$, where $e_{1}$, $e_{2}$ and so on are natural numbers. The expression "expr" may not contain free variables.

It does this by trying all possible combinations for $e_{1}$, $e_{2}$, ... 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.

##### Examples:
 In> RadSimp(Sqrt(9+4*Sqrt(2))) Out> Sqrt(8)+1; In> RadSimp(Sqrt(5+2*Sqrt(6)) \ +Sqrt(5-2*Sqrt(6))) Out> Sqrt(12); In> RadSimp(Sqrt(14+3*Sqrt(3+2 *Sqrt(5-12*Sqrt(3-2*Sqrt(2)))))) Out> Sqrt(2)+3; 

But this command may yield incorrect results:

 In> RadSimp(Sqrt(1+10^(-6))) Out> 1; 

Simplify , N .

### FactorialSimplify -- Simplify hypergeometric expressions containing factorials

##### Calling format:
 FactorialSimplify(expression) 

##### Parameters:
expression -- expression to simplify

##### Description:
FactorialSimplify takes an expression that may contain factorials, and tries to simplify it. An expression like $\frac{\left( n + 1\right) !}{n!}$ would simplify to $n + 1$.

The following steps are taken to simplify:

• binomials are expanded into factorials
• the expression is flattened as much as possible, to reduce it to a sum of simple rational terms
• expressions like
• $\frac{p ^{n}}{p ^{m}}$ are reduced to $p ^{n - m}$ if $n - m$ is an integer
• expressions like
• $\frac{n!}{m!}$ are simplified if $n - m$ is an integer

The function Simplify is used to determine if the relevant expressions $n - m$ are integers.

##### Example:
 In> FactorialSimplify( (n-k+1)! / (n-k)! ) Out> n+1-k In> FactorialSimplify(n! / Bin(n,k)) Out> k! *(n-k)! In> FactorialSimplify(2^(n+1)/2^n) Out> 2 

Simplify , ! , Bin .

### LnExpand -- expand a logarithmic expression using standard logarithm rules

##### Calling format:
 LnExpand(expr) 

##### Parameters:
expr -- the logarithm of an expression

##### Description:
LnExpand takes an expression of the form $\ln \mathrm{ expr }$, 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 $\ln x + 2 \ln 3$.

##### Example:
 In> LnExpand(Ln(a*b^n)) Out> Ln(a)+Ln(b)*n In> LnExpand(Ln(a^m/b^n)) Out> Ln(a)*m-Ln(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) 

Ln , LnCombine , Factors .

### LnCombine -- combine logarithmic expressions using standard logarithm rules

##### Calling format:
 LnCombine(expr) 

##### Parameters:
expr -- an expression possibly containing multiple Ln terms to be combined

##### Description:
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.

##### Example:
 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) 

Ln , LnExpand .

### TrigSimpCombine -- combine products of trigonometric functions

##### Calling format:
 TrigSimpCombine(expr) 

##### Parameters:
expr -- expression to simplify

##### Description:
This function applies the product rules of trigonometry, e.g. $\cos u \sin v = \frac{1}{2} \left( \sin \left( v - u\right) + \sin \left( v + u\right) \right)$. 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.

##### Examples:
 In> PrettyPrinter'Set("PrettyForm"); True In> TrigSimpCombine(Cos(a)^2+Sin(a)^2) 1 In> TrigSimpCombine(Cos(a)^2-Sin(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