Hello,

I agree with some of the comments above: "of course" is useful to point out that some step is trivial (e.g. direct consequence of the definition), as opposed to the rest of non-trivial parts of the proof. Sometimes, "of course" is useful just as an stylistic resource in the writing, to introduce and connect a sentence to the previous one. But it can be very frustrating for the reader if this step is non-trivial, even though the author claims it is.

I was curious about this question and decided to find some examples in the "mathematical literature", as the original poster suggested. I looked through "A Course in Arithmetic", by J-P. Serre (which many consider a very good writer of mathematics) and the expression "of course" appears exactly twice. In both cases, "of course" appears in a parenthetical remark:

1) (p.35) Corollary. - For two nondegenerate quadratic forms over $\mathbb{F}_q$ to be equivalent it is necessary and sufficient that they have same rank and same discriminant.
(Of course the discriminant is viewed as an element of the quotient group $\mathbb{F}_q^\ast/\mathbb{F}_q^{\ast 2}$.)

2) (p.73) Let $A$ be a subset of $P$ [$P$ is the set of prime numbers]. One says that $A$ has for density a real number $k$ when the ratio
$$ \left(\sum_{p\in A}\frac{1}{p^s} \right)/ \left(\log \frac{1}{s-1}\right)$$
tends to $k$ when $s\to 1$. (Of course, one then has $0 \leq k \leq 1$.)

In example (1), the way the corollary is stated, a remark is needed - but (i) it is clear from the context that this is what the author means, and (ii) it is typical in this context to consider discriminants only up to squares. Here I see this "of course" as a reminder of (ii).

Example (2) is trickier, as it is not immediately obvious that the limit of the expression as $s\to 1$ is between $0$ and $1$. But I do not interpret this "of course" as a "clearly" in this case, but rather a sort-of "do not worry, if you go back and check Cor 2 in p. 70, you can convince yourself that $0\leq k \leq 1$, and it makes sense to call this number a density".

Álvaro

PS: In "A Course in Arithmetic", the word "clearly" appears many times, while "obviously" was never used in the entire book.

of course, what is routine depends heavily on one's background. $\endgroup$ – Qiaochu Yuan Feb 23 '10 at 21:45