[Axiom-developer] expand sin (nx) and cos (nx)

[Francois Maltey]

Hello Martin, hello everybody !

I try to improve the expand function, 
I want to use it with my students in the chapter "trigonometric functions"

Today I don't change the package MANIP.spad, but I write *.input files.
I hope I can compile it without problem later inside the MANIP.spad

At the end of this input file you can try : 

expand1 (cos (3*x))+%i*expand1 (sin (3*x)) - (cos(x)+%i*sin(x))^3
expand1 (cos (a+b))
expand1 (sin (a-b))  -- doesn't work
expand1 (sin (b-a))  -- is right

So I have several problems :

// A //

My first function is a function which gives the name of the function :

With mupad I done :
op (3*x+5*y, 1)   -> _plus
op (3*x, 1)       -> _mult
op (x^3, 1)       -> _power
op (sin(3*x), 1)  -> sin
op (cos(3*x), 1)  -> cos

With axiom I try this function :

operator1 x == 
  (retractIfCan x)@Union(Kernel Expression Integer,"failed")

but it works only for Expression Integer, not Expression Complex Integer.

Is it possible to translate the mupad function : mainFct := x -> op (x, 1)


// B //

Why must I add some coerces  in this input file

// C //

What are the usual Expression ... type.

Can I be sure that coef in Integer, Fraction Integer, Complex Integer ?
Is it possible to have Float or DoubleFloat...

How can I test if variable coef in an Integer or not ? 

Thanks a lot if you can give me your advice !

Francois maltey. [I post at 17h UTC, 18h in France] 

------------------------------------------------------------------------

You can test it in an *.input file. 
error remains.
But I'd like to have information about dirty functions as operator1 
or silly coerce.

------------------------------------------------------------------------

operator1 x == 
  (retractIfCan x)@Union(Kernel Expression Integer,"failed")

expand1 (a : Expression Integer) : Expression Integer ==
  op := operator1 a
  op case "failed" => a 
  arg := first argument (op::Kernel Expression Integer) -- I must coerce
  is? (operator op, "cos"::Symbol) => expandcos arg
  is? (operator op, "sin"::Symbol) => expandsin arg
  a

sinnx (n,x) == eval (imag ((c + %i * s)^n), [c, s], [cos x, sin x])
cosnx (n,x) == eval (real ((c + %i * s)^n), [c, s], [cos x, sin x])

expandsin arg ==
  arg = 0 => 0
  num := numer arg
  den := denom arg 
  head := leadingMonomial num 
  coef := leadingCoefficient num
  a := head / coef / den
  tail := reductum num 
  b := (tail / den)::Expression Integer               -- I must coerce 
  sinnx (coef, a) * expand1 (cos b) + cosnx (coef, a) * expand1 (sin b)

expandcos arg ==
  arg = 0 => 1
  num := numer arg
  den := denom arg 
  head := leadingMonomial num 
  coef := leadingCoefficient num
  a := head / coef / den
  tail := reductum num 
  b := (tail / den)::Expression Integer                -- I must coerce 
  cosnx (coef, a) * expand1 (cos b) - sinnx (coef, a) * expand1 (sin b)


-- You can try : 
expand1 (cos (3*x))+%i*expand1 (sin (3*x)) - (cos(x)+%i*sin(x))^3
expand1 (sin (a-b))
expand1 (sin (a+b))
expand1 (cos (b-a))