# Proper treatment of roots and powers in Series?

I have the following problem in Mathematica 9 on Linux. I let Mathematica compute the Series expansion:

Block[{$Assumptions = {r, p}\[Element] Reals}, Series[Log[r^2 - Sqrt[r^4 + p^2]], {p, 0, 0}] ]  But the output is a rather disappointing naked infinity: Log[r^2 - Sqrt[r^4]] + O[p]^1  Similarly, for: Block[{$Assumptions = {p, r} \[Element] Reals},
Series[1/(r^2 - Sqrt[r^4 + p^2]), {p, 0, 0}]
]


I get:

1/(r^2 - Sqrt[r^4]) + O[p]^1


Why is Mathematica handing me naked infinities like that, even though I specified that the computation is to be carried out in the real numbers? What do I do wrong? How can I avoid this?

EDIT:

To make my point more clear, I would expect the Series to give results like $$\log(c_1 p^2)+O(p^1)$$ and $$\frac{1}{c_2 p^2}+O(p^1)$$ as a regulated infinity, instead of $\log(0)$ and $\frac{1}{0}$.

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Are you disappointed about the output or do you think it's not correct ? – b.gatessucks Mar 5 '13 at 11:17
I am disappointed, since the Series function should give a regulated infinity, like $\frac{1}{p^2}$ or something, instead of a mathematically nonsensical $\frac{1}{0}$. – Kagaratsch Mar 5 '13 at 11:19
I have been told, that as a quick solution one can substitute r by ie Zeta[5]. Any zeta of odd numbers greater than 1 will do. After the expansion is done one simply has to substitute back. – Kagaratsch Mar 5 '13 at 14:11
Not a full-fledged answer, Mathematica's behaviour still puzzles me, but something that may point you in the right direction: you do get the expected result if you assume r>0 && p>0. The assumption r<0 && p>0 works too, but (r<0 || r>0) && p>0 does not... – Marcks Thomas Mar 5 '13 at 14:42