Consider the following:
var("p, Np, w")
Vsp = function("Vsp",p)
VSm(m) = sum(Vsp(p=p)*exp(I*w*p*m),p,1,Np-1)
def KS(Sm): return sum(Sm*exp(-I*w*p*m),m,0,Np-1)
KS(VSm(m))
--> sum(sum(Vsp(I)*e^(-I*m*p*w - m*w), I, 1, Np - 1), m, 0, Np - 1)
This is incorrect. Sage has replaced the dummy sum variable "p" with "I",
which conflicts with the imaginary number "I". This can be verified by
using "j" instead of "I":
var("p, Np, w, j")
Vsp = function("Vsp",p)
VSm(m) = sum(Vsp(p=p)*exp(j*w*p*m),p,1,Np-1)
def KS(Sm): return sum(Sm*exp(-j*w*p*m),m,0,Np-1)
KS(VSm(m))
--> sum(sum(Vsp(I)*e^(-j*m*p*w + I*j*m*w), I, 1, Np - 1), m, 0, Np - 1)
Note incorrect term "-m*w" in previous expression has become "I*j*m*w".
A workaround is to use an unused variable in the original sum, so that Sage
doesn't replace it:
var("p, Np, w, q")
Vsp = function("Vsp",p)
VSm(m) = sum(Vsp(p=q)*exp(I*w*q*m),q,1,Np-1)
def KS(Sm): return sum(Sm*exp(-I*w*p*m),m,0,Np-1)
KS(VSm(m))
--> sum(sum(Vsp(q)*e^(-I*m*p*w + I*m*q*w), q, 1, Np - 1), m, 0, Np - 1)
which is the desired result.
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