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There's a residue method on symbolic expressions which is documented to do the thing you want, but of course you should check that it does indeed do what you want.

I think you can use:

k=3
br(x)=sqrt(x)-sqrt(1/x)
q1,q2=var('q1,q2')
X=var('x_0,x_1,x_2')
rho=[1,q1,q2]
chi=1/prod([ br(X[j])*br(q1/X[j])*X[j]*br(q2/X[j]) for j in range(k)])
[chi.residue(xi==rhoi) for xi,rhoi in zip(X,rho)]

There's a residue method on symbolic expressions which is documented to do the thing you want, but of course you should check that it does indeed do what you want.

I think you can use:

k=3
br(x)=sqrt(x)-sqrt(1/x)
q1,q2=var('q1,q2')
X=var('x_0,x_1,x_2')
rho=[1,q1,q2]
chi=1/prod([ br(X[j])*br(q1/X[j])*X[j]*br(q2/X[j]) for j in range(k)])
[chi.residue(xi==rhoi) for xi,rhoi in zip(X,rho)]

for xi,rhoi in zip(X,rho):
    chi = chi.residue(xi,rhoi)
print chi