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SandBoxSinCosRules

Edit detail for SandBoxSinCosRules revision 1 of 2

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Editor: Bill Page Time: 2009/10/16 19:40:14 GMT-7
Note: split

changed:
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Sin and Cos Rules

\begin{axiom}
x:=v1*sin(p1)+v2*sin(p2)
y:= a0 + a1*x + a2*x^2 + a3*x^3
sinCosProducts := rule
  sin(x)*sin(y) == (cos(x-y) - cos(x+y))/2
  cos(x)*cos(y) == (cos(x-y) + cos(x+y))/2
  sin(x)*cos(y) == (sin(x-y) + sin(x+y))/2
  sin(x)^2 == (1 - cos(2*x))/2
  sin(x)^3 == sin(x)*(1 - cos(2*x))/2

sinCosProducts(y)
\end{axiom}

Integration

\begin{axiom}
)clear completely
integrate(%e ^(-x*x), x=%minusInfinity..%plusInfinity)
\end{axiom}

\begin{axiom}
integrate(integrate(x+y, x),y)
\end{axiom}

\begin{axiom}
integrate(exp(-b*h*sin(o))*sin(o),o=0..2*%pi)
\end{axiom}

Gosper's algorithm is implemented in Axiom

\begin{axiom}
p:Polynomial Integer:=1+x+x^2
sum(p,x=0..n)
e:=binomial(2*k,k)/4^k
sum(e,k=0..n)
sum(%,n=0..n)
factor(4*n^2+8*n+3)
\end{axiom}
Examples from Petkovsek, Wilf, Zeilberger

\begin{reduce}
exp(log(x)+log(y^-1));
\end{reduce}


Symbols, kernels, variables, expressions ... difficult to understand

\begin{axiom}
e0:Expression Integer:=1+2*x^2+x
kernels(e0)
p0:=e0::(Polynomial Integer)
variables(p0)
e1:Expression Integer:=x*sin(t)/cos(t)+1
kernels(e1)
solve(e1=0,x)
e2:UP(x,Expression Integer):=sin(t)/cos(t)*x
\end{axiom}


Expressions and substitution

\begin{axiom}
f:=operator 'f; e:=1+a*x**2+f(y)*x^3; eq:=[f(y)=r]; peq:=subst(e,eq)::(Polynomial Integer)=0;sol:=radicalSolve(peq,x);
--[(lhs(s)=subst(rhs(s),r=f(y))) for s in sol]
--As a list there is no output shown
--x1=subst (rhs(sol.1),r=f(y))
--x2=subst (rhs(sol.2),r=f(y))
x3=subst (rhs(sol.3),r=f(y))
\end{axiom}


Parsing expressions ???

\begin{axiom}
e:Expression Integer:=x*y
isMult(e)
e1:Expression Integer:=x
isMult(e1)
isMult(e1*e)
\end{axiom}

Sin and Cos Rules

axiom

x:=v1*sin(p1)+v2*sin(p2)

$\label{eq1}{v 2 \ {\sin \left({p 2}\right)}}+{v 1 \ {\sin \left({p 1}\right)}}$

(1)

Type: Expression(Integer)

axiom

y:= a0 + a1*x + a2*x^2 + a3*x^3

$\label{eq2}\begin{array}{@{}l} \displaystyle {a 3 \ {v 2^3}\ {{\sin \left({p 2}\right)}^3}}+{{\left({3 \ a 3 \ v 1 \ {v 2^2}\ {\sin \left({p 1}\right)}}+{a 2 \ {v 2^2}}\right)}\ {{\sin \left({p 2}\right)}^2}}+ \ \ \displaystyle {{\left({3 \ a 3 \ {v 1^2}\ v 2 \ {{\sin \left({p 1}\right)}^2}}+{2 \ a 2 \ v 1 \ v 2 \ {\sin \left({p 1}\right)}}+{a 1 \ v 2}\right)}\ {\sin \left({p 2}\right)}}+ \ \ \displaystyle {a 3 \ {v 1^3}\ {{\sin \left({p 1}\right)}^3}}+{a 2 \ {v 1^2}\ {{\sin \left({p 1}\right)}^2}}+{a 1 \ v 1 \ {\sin \left({p 1}\right)}}+ a 0$

(2)

Type: Expression(Integer)

axiom

sinCosProducts := rule
  sin(x)*sin(y) == (cos(x-y) - cos(x+y))/2
  cos(x)*cos(y) == (cos(x-y) + cos(x+y))/2
  sin(x)*cos(y) == (sin(x-y) + sin(x+y))/2
  sin(x)^2 == (1 - cos(2*x))/2
  sin(x)^3 == sin(x)*(1 - cos(2*x))/2

$\label{eq3}\begin{array}{@{}l} \displaystyle \left\{{{\%B \ {\sin \left({x}\right)}\ {\sin \left({y}\right)}}\mbox{\rm = =}{{-{\%B \ {\cos \left({y + x}\right)}}+{\%B \ {\cos \left({y - x}\right)}}}\over 2}}, \: \right. \ \ \displaystyle \left.{{\%C \ {\cos \left({x}\right)}\ {\cos \left({y}\right)}}\mbox{\rm = =}{{{\%C \ {\cos \left({y + x}\right)}}+{\%C \ {\cos \left({y - x}\right)}}}\over 2}}, \: \right. \ \ \displaystyle \left.{{\%D \ {\cos \left({y}\right)}\ {\sin \left({x}\right)}}\mbox{\rm = =}{{{\%D \ {\sin \left({y + x}\right)}}-{\%D \ {\sin \left({y - x}\right)}}}\over 2}}, \: \right. \ \ \displaystyle \left.{{{\sin \left({x}\right)}^2}\mbox{\rm = =}{{-{\cos \left({2 \ x}\right)}+ 1}\over 2}}, \: \right. \ \ \displaystyle \left.{{{\sin \left({x}\right)}^3}\mbox{\rm = =}{{{\left(-{\cos \left({2 \ x}\right)}+ 1 \right)}\ {\sin \left({x}\right)}}\over 2}}\right\}$

(3)

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

axiom

sinCosProducts(y)

$\label{eq4}{\left( \begin{array}{@{}l} \displaystyle -{a 3 \ {v 2^3}\ {\sin \left({3 \ p 2}\right)}}-{3 \ a 3 \ v 1 \ {v 2^2}\ {\sin \left({{2 \ p 2}+ p 1}\right)}}+ \ \ \displaystyle {3 \ a 3 \ v 1 \ {v 2^2}\ {\sin \left({{2 \ p 2}- p 1}\right)}}- \ \ \displaystyle {3 \ a 3 \ {v 1^2}\ v 2 \ {\sin \left({p 2 +{2 \ p 1}}\right)}}+ \ \ \displaystyle {{\left({3 \ a 3 \ {v 2^3}}+{{\left({6 \ a 3 \ {v 1^2}}+{4 \ a 1}\right)}\ v 2}\right)}\ {\sin \left({p 2}\right)}}- \ \ \displaystyle {3 \ a 3 \ {v 1^2}\ v 2 \ {\sin \left({p 2 -{2 \ p 1}}\right)}}-{a 3 \ {v 1^3}\ {\sin \left({3 \ p 1}\right)}}+ \ \ \displaystyle {{\left({6 \ a 3 \ v 1 \ {v 2^2}}+{3 \ a 3 \ {v 1^3}}+{4 \ a 1 \ v 1}\right)}\ {\sin \left({p 1}\right)}}- \ \ \displaystyle {2 \ a 2 \ {v 2^2}\ {\cos \left({2 \ p 2}\right)}}-{4 \ a 2 \ v 1 \ v 2 \ {\cos \left({p 2 + p 1}\right)}}+ \ \ \displaystyle {4 \ a 2 \ v 1 \ v 2 \ {\cos \left({p 2 - p 1}\right)}}-{2 \ a 2 \ {v 1^2}\ {\cos \left({2 \ p 1}\right)}}+ \ \ \displaystyle {2 \ a 2 \ {v 2^2}}+{2 \ a 2 \ {v 1^2}}+{4 \ a 0}$

(4)

Type: Expression(Integer)

Integration

axiom

)clear completely

   All user variables and function definitions have been cleared.
   All )browse facility databases have been cleared.
   Internally cached functions and constructors have been cleared.
   )clear completely is finished.
integrate(%e ^(-x*x), x=%minusInfinity..%plusInfinity)

$\label{eq5}\sqrt{\pi}$

(5)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

axiom

integrate(integrate(x+y, x),y)

$\label{eq6}{{1 \over 2}\ x \ {y^2}}+{{1 \over 2}\ {x^2}\ y}$

(6)

Type: Polynomial(Fraction(Integer))

axiom

integrate(exp(-b*h*sin(o))*sin(o),o=0..2*%pi)

$\label{eq7}\mbox{\tt "failed"}$

(7)

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

Gosper's algorithm is implemented in Axiom

axiom

p:Polynomial Integer:=1+x+x^2

$\label{eq8}{x^2}+ x + 1$

(8)

Type: Polynomial(Integer)

axiom

sum(p,x=0..n)

$\label{eq9}{{n^3}+{3 \ {n^2}}+{5 \ n}+ 3}\over 3$

(9)

Type: Fraction(Polynomial(Integer))

axiom

e:=binomial(2*k,k)/4^k

$\label{eq10}{\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ k}, \: k}\right)}\over{4^k}$

(10)

Type: Expression(Integer)

axiom

sum(e,k=0..n)

$\label{eq11}{{\left({2 \ n}+ 1 \right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ n}, \: n}\right)}}\over{4^n}$

(11)

Type: Expression(Integer)

axiom

sum(%,n=0..n)

$\label{eq12}{{\left({4 \ {n^2}}+{8 \ n}+ 3 \right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ n}, \: n}\right)}}\over{3 \ {4^n}}$

(12)

Type: Expression(Integer)

axiom

factor(4*n^2+8*n+3)

$\label{eq13}{\left({2 \ n}+ 1 \right)}\ {\left({2 \ n}+ 3 \right)}$

(13)

Type: Factored(Polynomial(Integer))

Examples from Petkovsek, Wilf, Zeilberger

exp(log(x)+log(y^-1));

reduce

$\displaylines{\qdd \frac{x}{ y} \cr}$

Symbols, kernels, variables, expressions ... difficult to understand

axiom

e0:Expression Integer:=1+2*x^2+x

$\label{eq14}{2 \ {x^2}}+ x + 1$

(14)

Type: Expression(Integer)

axiom

kernels(e0)

$\label{eq15}\left[ x \right]$

(15)

Type: List(Kernel(Expression(Integer)))

axiom

p0:=e0::(Polynomial Integer)

$\label{eq16}{2 \ {x^2}}+ x + 1$

(16)

Type: Polynomial(Integer)

axiom

variables(p0)

$\label{eq17}\left[ x \right]$

(17)

Type: List(Symbol)

axiom

e1:Expression Integer:=x*sin(t)/cos(t)+1

$\label{eq18}{{x \ {\sin \left({t}\right)}}+{\cos \left({t}\right)}}\over{\cos \left({t}\right)}$

(18)

Type: Expression(Integer)

axiom

kernels(e1)

$\label{eq19}\left[{\sin \left({t}\right)}, \:{\cos \left({t}\right)}, \: x \right]$

(19)

Type: List(Kernel(Expression(Integer)))

axiom

solve(e1=0,x)

$\label{eq20}\left[{x = -{{\cos \left({t}\right)}\over{\sin \left({t}\right)}}}\right]$

(20)

Type: List(Equation(Expression(Integer)))

axiom

e2:UP(x,Expression Integer):=sin(t)/cos(t)*x

$\label{eq21}{{\sin \left({t}\right)}\over{\cos \left({t}\right)}}\ x$

(21)

Type: UnivariatePolynomial?(x,Expression(Integer))

Expressions and substitution

axiom

f:=operator 'f; e:=1+a*x**2+f(y)*x^3; eq:=[f(y)=r]; peq:=subst(e,eq)::(Polynomial Integer)=0;sol:=radicalSolve(peq,x);

Type: List(Equation(Expression(Integer)))

axiom

--[(lhs(s)=subst(rhs(s),r=f(y))) for s in sol]
--As a list there is no output shown
--x1=subst (rhs(sol.1),r=f(y))
--x2=subst (rhs(sol.2),r=f(y))
x3=subst (rhs(sol.3),r=f(y))

$\label{eq22}\begin{array}{@{}l} \displaystyle x 3 ={{\left( \begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle 9 \ {{f \left({y}\right)}^2}\ \cdot \ \ \displaystyle {{\root{3}\of{{{{54}\ {{f \left({y}\right)}^3}\ {\sqrt{{{{27}\ {{f \left({y}\right)}^2}}+{4 \ {a^3}}}\over{{108}\ {{f \left({y}\right)}^4}}}}}-{{27}\ {{f \left({y}\right)}^2}}-{2 \ {a^3}}}\over{{54}\ {{f \left({y}\right)}^3}}}}^2}$