> If I simply define and use a rule
>
> exp_to_bessel_rule := rule exp( sqrt(-1) * z * sin(t) ) == besselJ( N, z
> ) * exp( sqrt( -1 ) * N * t )
> exp_to_bessel_rule( test_eqn )
>
> then I'll get expression with 2 identical summation indices ( 'N' in this
> case ), which is clearly not something I want.
Right. That's clear. The problem is that your besselJ(N, ...) is only a
workaround to some other problem.
You want sum_{N=-\infty}^\infty foo(N) as a result. N is bound.
As Waldek said, you have to invent your own operator for the infinite sum.
I experimented with
mysum := operator 'mysum
mysum(minusInf, plusInf, N +-> besselJ(N, a) * exp( %i * N * t ))
to represent your sum. Unfortunately +-> cannot be used in that place.
Maybe the following session might help you. Maybe it's still not what
you want, since you are left with mysum expressions that you have to
interpret yourself (by adding even more rules to the system).
Anyway, I think what is below is in the spirit of what Waldek suggested.
Ralf
(1) -> )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.
(1) -> mysum := operator 'mysum
(1) mysum
Type:
BasicOperator
(2) -> mi -- represents minusInfinity
(2) mi
Type:
Variable(mi)
(3) -> pi -- represents plusInfinity
(3) pi
Type:
Variable(pi)
(5) -> B(N, l, h, a, t) == mysum(l, h, N, besselJ(N, a) * exp(sqrt(-1) *
N * t ))
Compiled code for bess has been cleared.
Type:
Void
(6) -> bess(a,t) == (N:Symbol:=symbol GENSYM()$Lisp;B(N,mi,pi,a,t))
1 old definition(s) deleted for function or rule bess
Type:
Void
(7) -> bess(a,t)
Compiling function B with type (Symbol,Variable(mi),Variable(pi),
Variable(a),Variable(t)) -> Expression(Integer)
Compiling function bess with type (Variable(a),Variable(t)) ->
Expression(Integer)
+---+
G658 t\|- 1
(7) mysum(mi,pi,G658,%e besselJ(G658,a))
Type:
Expression(Integer)
(8) -> bess(a,t)
+---+
G659 t\|- 1
(8) mysum(mi,pi,G659,%e besselJ(G659,a))
Type:
Expression(Integer)
(9) -> test_eqn := exp(sqrt(-1)*a*sin(omg*t))*exp(sqrt(-1)*b*sin(2*omg*t))
+---+ +---+
a\|- 1 sin(omg t) b\|- 1 sin(2omg t)
(9) %e %e
Type:
Expression(Integer)
(10) -> exp_to_bessel_rule := rule exp(sqrt(-1)*z*sin(t)) == bess(z,t)
+---+
z\|- 1 sin(t)
(10) %e == 'bess(z,t)
Type:
RewriteRule(Integer,Integer,Expression(Integer))
(11) -> exp_to_bessel_rule( test_eqn )
Compiling function B with type (Symbol,Variable(mi),Variable(pi),
Variable(b),Polynomial(Integer)) -> Expression(Integer)
Compiling function bess with type (Variable(b),Polynomial(Integer))
-> Expression(Integer)
Compiling function B with type (Symbol,Variable(mi),Variable(pi),
Variable(a),Polynomial(Integer)) -> Expression(Integer)
Compiling function bess with type (Variable(a),Polynomial(Integer))
-> Expression(Integer)
(11)
+---+
2G688 omg t\|- 1
mysum(mi,pi,G688,%e besselJ(G688,b))
*
+---+
G691 omg t\|- 1
mysum(mi,pi,G691,%e besselJ(G691,a))
Type:
Expression(Integer)
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