series of x^s, when s is symbolic

[dkrenn]

We have the following behaviour:

sage: var('s')
s
sage: (x^s).series(x, 0)
Order(1)
sage: (x^s).series(x, 1)
(0^s) + Order(x)
sage: (x^s).series(x, 2)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-4-5abc79662303> in <module>()
----> 1 (x**s).series(x, Integer(2))

/usr/opt/sage-6.1.1/local/lib/python2.7/site-packages/sage/symbolic/expression.so in sage.symbolic.expression.Expression.series (sage/symbolic/expression.cpp:17596)()

ValueError: power::eval(): division by zero

This output is weird and definitely wrong (since the correct output depends strongly on s.).

CC: @mforets

Component: symbolics

Keywords: symbolic, series, exponent

Issue created by migration from https://trac.sagemath.org/ticket/15854

[dkrenn]

comment:3

Still there in 6.6

[rwst]

comment:4

Could you please specify what output exactly to expect?

[mforets]

comment:5

there is no error if you declare s as integer:

sage: s = SR.var('s', domain='integer')
sage: (x^s).series(x, 2) # ok (?) or we expect x^s
(0^s) + (0^(s - 1)*s)*x + Order(x^2)

to compare, W|A gives various series representations. in this case maybe it could answer just x^s?

however, there is this closely related issue:

sage: n = SR.var('n', domain='integer')
sage: ((x+1/x)**n).series(x)  # wrong (what happened with n?)
1 + Order(x^20)

one expects something like 1/x^n*(1 + n*x^2 + O(x^3)).

this behaviour has side effects for example if you want to compute the residue of this function:

sage: f = 1/x*((x^2+1)/(2*x))**(2*k)
sage: f.residue(x==0)   # wrong
(1/2)^(2*k)

in fact:

sage: f(k=4)
1/256*(x^2 + 1)^8/x^9
sage: f(k=4).residue(x==0)
35/128
sage: f.residue(x==0).subs(k==4)
1/256
sage: res(k) = 1/2**(2*k)*binomial(2*k, k)   # correct answer
sage: res(k=4)
35/128

a bit unrelated, but let me mention that SymPy's residue gives wrong result for this one (0), and giac gives unevaluated expression.