After quite some searching I did not succeed to find documentation for sage
functions to work with complex numbers as much as I would like.
For example if I have a complicated rational expression, how can I tell
Sage "bring this to the form a + bi". It seems real() and imag() only
work
if no pre-processing is needed. How about "multiply numerator and
denominator by denominator.conjugate()" ? There's probably a chapter in
the documentation about this, could someone please point me to it, I seem
to be incompetent at finding it, sorry.
Since people want something concrete to look at, not just a general
question, here is some code. You'll see that it computes a certain
complex function (actually two of them)
with integer parameters N and M, the solution(s) of a certain equation.
I'd like to compute that the absolute value of those expressions must be 1.
The
code below computes it numerically for some more or less random values of
N and M, and it is 1.0000 for those values, but I can't figure out how to
compute it symbolically. Also, if there's a better way to do polynomial
division than I've used below, please tell me.
def nov13b():
var('p,q,r,N,M,x')
a = sqrt(3)/2
b = (x-x^(-1))/(2*i)
c = (sqrt(3)/2)* (x+x^(-1))/2 + (1/2)*(x-x^(-1))/(2*i)
X = (M/3)*(a+b+c)
f = 24*(X^2-N*b*c)*x^2
g = (f.maxima_methods().divide(x+1)[0]).full_simplify()
print(g.full_simplify())
print("")
t = exp(-pi*i/3)
print(g(x=t).full_simplify())
print("")
h = (g.maxima_methods().divide(x-t)[0]).full_simplify()
print("h = ")
print(h)
print("")
answers = solve(h,x)
assume(N,'integer')
assume(M,'integer')
for u in answers:
print("")
ans = u.rhs().simplify()
for k in range(230,245):
ans_numerical = abs(ans.substitute(M=11,N=243)).simplify()
print(n(ans_numerical))
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