MathAction Manipulating Expressions

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SandBoxDomainOfComputation

The domain InputForm can be quite useful for manipulating parts of expressions. For example

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(1) -> ex1:=integrate(log(x)+x, x)

$\label{eq1}\frac{{2 \ x \ {\log \left({x}\right)}}+{{x}^{2}}-{2 \ x}}{2}$

(1)

Type: Union(Expression(Integer),...)

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)set output tex off

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)set output algebra on

%::InputForm

   (2)  (/ (+ (* (* 2 x) (log x)) (+ (^ x 2) (* - 2 x))) 2)

Type: InputForm

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)set output tex on

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)set output algebra off

ex2:=interpret((%::InputForm).2.2)

$\label{eq2}2 \ x \ {\log \left({x}\right)}$

(2)

Type: Expression(Integer)

If you would like to do this with a more common type of expression and hide the details, you can define

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op(n,x) == interpret((x::InputForm).(n+1))

Type: Void

Then manipulating expressions looks like this:

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op(1,ex1)

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Compiling function op with type (PositiveInteger, Expression(Integer
      )) -> Any

$\label{eq3}{2 \ x \ {\log \left({x}\right)}}+{{x}^{2}}-{2 \ x}$

(3)

Type: Expression(Integer)

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op(1,%)

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Compiling function op with type (PositiveInteger, Any) -> Any

$\label{eq4}2 \ x \ {\log \left({x}\right)}$

(4)

Type: Expression(Integer)

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(op(1,op(1,ex1))-op(2,op(1,ex1)))/op(2,ex1)

$\label{eq5}\frac{{2 \ x \ {\log \left({x}\right)}}-{{x}^{2}}+{2 \ x}}{2}$

(5)

Type: Expression(Integer)

Rules and Pattern Matching (from WesterProblemSet)

Trigonometric manipulations---these are typically difficult for students

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r:= cos(3*x)/cos(x)

$\label{eq6}\frac{\cos \left({3 \ x}\right)}{\cos \left({x}\right)}$

(6)

Type: Expression(Integer)

=> cos(x)^2 - 3 sin(x)^2 or similar

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real(complexNormalize(r))

$\label{eq7}-{2 \ {{\sin \left({x}\right)}^{2}}}+{2 \ {{\cos \left({x}\right)}^{2}}}- 1$

(7)

Type: Expression(Integer)

=> 2 cos(2 x) - 1

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real(normalize(simplify(complexNormalize(r))))

$\label{eq8}{2 \ {\cos \left({2 \ x}\right)}}- 1$

(8)

Type: Expression(Integer)

Use rewrite rules => cos(x)^2 - 3 sin(x)^2

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sincosAngles:= rule
   cos((n | integer?(n)) * x) ==
      cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x)
   sin((n | integer?(n)) * x) ==
      sin((n - 1)*x) * cos(x) + cos((n - 1)*x) * sin(x)

$\label{eq9}\begin{array}{@{}l} \displaystyle \left\{{= = \left({{\cos \left({n \ x}\right)}, \:{-{{\sin \left({x}\right)}\ {\sin \left({{\left(n - 1 \right)}\ x}\right)}}+{{\cos \left({x}\right)}\ {\cos \left({{\left(n - 1 \right)}\ x}\right)}}}}\right)}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{= = \left({{\sin \left({n \ x}\right)}, \:{{{\cos \left({x}\right)}\ {\sin \left({{\left(n - 1 \right)}\ x}\right)}}+{{\cos \left({{\left(n - 1 \right)}\ x}\right)}\ {\sin \left({x}\right)}}}}\right)}\right\}$

(9)

Type: Ruleset(Integer,Integer,Expression(Integer))

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sincosAngles r

$\label{eq10}-{3 \ {{\sin \left({x}\right)}^{2}}}+{{\cos \left({x}\right)}^{2}}$

(10)

Type: Expression(Integer)

Other Operations

The domain FunctionSpace? includes the following operations:

    isExpt(p,f:Symbol) returns [x, n] if p = x^n and n <> 0 and x = f(a)
    isExpt(p,op:BasicOperator) returns [x, n] if p = x^n and n <> 0 and x = op(a)
    isExpt(p) returns [x, n] if p = x^n and n <> 0
    isMult(p) returns [n, x] if p = n * x and n <> 0
    isPlus(p) returns [m1,...,mn] if p = m1 +...+ mn and n > 1
    isPower(p) returns [x, n] if p = x^n and n <> 0
    isTimes(p) returns [a1,...,an] if p = a1*...*an and n > 1

If these conditions are not met, then the above operations return "failed".

For example,

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isMult(3*x)

$\label{eq11}\left[{coef = 3}, \:{var = x}\right]$

(11)

Type: Union(Record(coef: Integer,var: Kernel(Expression(Integer))),...)

but

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isMult(x*y)

$\label{eq12}\verb#"failed"#$

(12)

Type: Union("failed",...)

In the context of Expression Integer, or Polynomial Integer the parameter n must be an Integer. The Symbol y is not an Integer.

Not exactly analogously

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isPower(x^y)

$\label{eq13}\left[{val ={{x}^{y}}}, \:{exponent = 1}\right]$

(13)

Type: Union(Record(val: Expression(Integer),exponent: Integer),...)

whereas

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isPower(x^10)

$\label{eq14}\left[{val = x}, \:{exponent ={10}}\right]$

(14)

Type: Union(Record(val: Expression(Integer),exponent: Integer),...)

In the first case the Integer is assume to be 1.

We have:

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isTimes(x*y*z)

$\label{eq15}\left[ z , \: y , \: x \right]$

(15)

Type: Union(List(Polynomial(Integer)),...)

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isPlus(x+y+z*y)

$\label{eq16}\left[{y \ z}, \: y , \: x \right]$

(16)

Type: Union(List(Polynomial(Integer)),...)

Whereas

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isTimes((x+y)*z)

$\label{eq17}\verb#"failed"#$

(17)

Type: Union("failed",...)

That is because the expression is internally treated as a MultivariatePolynomial like this:

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((x+y)*z)::MPOLY([x,y,z],INT)

$\label{eq18}{z \ x}+{z \ y}$

(18)

Type: MultivariatePolynomial?([x,y,z],Integer)

If you say:

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isPlus((x+y)*z)

$\label{eq19}\left[{y \ z}, \:{x \ z}\right]$

(19)

Type: Union(List(Polynomial(Integer)),...)

perhaps the result makes sense?

For some of the details of these operations I consulted the actual algebra code at:

http://axiom-wiki.newsynthesis.org/axiom--test--1/src/algebra/FspaceSpad

Click on pdf or dvi to see the documentation.

You can also enter expressions like isTimes in the search box on the upper right and see all the places in the algebra where this operation is defined and used.

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