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First of all, here is the answer, since right now only 10K+ users can see it. (The original had a plot in it, which I haven't copied.)
You can use this formula to solve your equation:
$$f^{[1/2]}(x) = \sum_{m=0}^{\infty} \binom{1/2}{m} \sum_{k=0}^m \binom{m}{n} (-1)^{m-k} f^{[k]}(x)$$
where $f^{[k]}(x)$ is a $k$-th iterate of $x$. Here is a plot for $\sin x$.
But for cosine this method apparently doesn't work.
I would not have deleted this answer, were I a moderator. But it falls into the frustrating category, which I know very well from grading exams, of the answer which mentions some useful knowledge but does not show how to actually apply it to the problem, or an awareness of when it can be applied.
It is interesting to know that this formula exists, and it is new to me. So I think there is some valuable information in this answer.
The obvious problem on looking at this formula is that it is not clear when it converges, either in the sense of analysis or in the ring of formal power series. Even if $f(x)=0$, computing any term of $f^{[1/2]}$ appears to require evaluating an infinite sum. If $f(0)=0$, and $|f'(0)|<1$, then I'm pretty sure I could show the sum converges (warning, I have not checked all the details here), but when $f'(1)=1$ even that isn't clear to me, so I don't know whether your formula is even meaningful for the case of sine where you compute it.
Ironically, if you (Anixx) had thought about these issues, you would see that the case of cosine is less subtle. Since cosine has a fixed point at approximately 0.739, where the derivative is clearly small, you could just shift your power series to be centered around this point and obtain a nice sum which would probably converge quickly. This would build nicely on Gerald Edgar's answer, which uses the same idea of shifting to a fixed point, but does not obtain an explicit solution.
As I said, I would not have deleted this answer, because it contributes valuable information. But I think it generally suffers from a willingness to thinking about mathematics as formulas, rather than thinking about when those formulas are meaningful.
As an example of the subtleties which can arise, I was recently reminded of http://www.ams.org/mathscinet-getitem?mr=97532 , which shows that there is no analytic function f, convergent in a neighborhood of 0, such that f(f(z))=e^z-1. This despite the fact that there is a perfectly good formal power series solution.
Most mathematicians take the view that, when preparing a communication, one should anticipate the arguments that will arise, and address them proactively. This is why math papers are hard to read, but also why one mathematician can write a paper which is useful to thousands.
To me, there is a big difference between what I will criticize in a comment, and what I would vote to delete. I wouldn't delete this, but I would have left comments asking about the points I raised above (and, also, asking for a citation for this result). MO software allows for collaborative editing and incremental improvement, and I think we should take advantage of that. But I also try to write answers which are as thorough and clear as I can make them before revision. When someone seems to be paying no attention to these concerns, I am not surprised that it annoys other mathematicians.
@Anixx: You asked why your question got deleted, and Ryan has given you what to me sounds like a very plausible explanation -- there's clearly something undesirable about the answer (hence the 4 downvotes), and discussing with the matter with you has proven in the past to be too burdensome to be worth the effort. Your discussion with David is only furthering Ryan's point in this regard. If you take the time to include some sort of theoretical framework (proof of convergence, references, even a heuristic argument, etc.), rather than simply an isolated claim, and proactively discuss potential issues that arise with your answer, I think you will find that you are met with much less resistance.
I'm going to regret this, but I've reposted your answer, because I don't think it should have been deleted, merely criticized.
Now, can I make some points?
(1) Note that my rewrite actually addresses the relevant context, and states which things are theorems and which ones only sometimes work. It provides a useful link to Wikipedia, where the reader can start finding references to the literature. This wasn't that hard for me, all I had to do was to google Newton Series and skim a few screens of text. It would have been much easier for you.
You definitely seem to know a lot of mathematics. But you also need to learn how to present it so it is useful to other mathematicians.
(2) It is rude to rely on others to do this sort of work. Now, it is not as rude on the web as it is in print. Here, once one person does this effort, everyone can benefit from it. But it still should be the obligation of the original author to clean up their work as much as they can, rather than relying on their readers to do so.
I have a stack of exams to grade today. When I do that, I will go through every problem as carefully as I can, try to work out what is meant by ambiguous sentences, and do my best to suggest more precise phrasings. This is important work, and I try to do it well. When you're not paying me, I shouldn't have to do that. After I grade that stack of exams, I have a paper to edit. I will be going through that paper trying to make every point as precise as possible, so that no one else has to guess what I meant.
(3) In light of the stack of the exams, and the paper, I'm bowing out of this thread.
I was the person who deleted the answer. It was problematic for a few reasons:
I see that Anixx is still claiming to have provided an "exact and general answer" and a "single general formula" when the formula completely fails to answer the question that was given. In modern mathematics, our norms of discussion dictate that when we provide a formula as a solution, we must present a convincing argument that the formula is well-defined and solves the problem. If no such argument is given, then the formula is not a complete solution. The answers that Anixx gave may have been acceptable 250 years ago, but now we know that an explicit formula can fail badly when we leave its domain of applicability.
I think I was wrong to delete Anixx's roughly identical answer to Kevin Buzzard's question, since that is where the formula belongs (together with a suitable discussion of convergence). I see that David Speyer has reposted a cleaner version.
I agree with David Speyer's decision to repost Annix's answer, and I appreciate his additions to the answer, which help to explain it. I don't think that these answers should be deleted simply because they don't demonstrate convergence. Annix's answers have certainly taught me about a method and some phenomena of which I was previously unaware.
If people don't want to engage with Annix, they are welcome not to. But as far as I am aware, he is not spamming the site; rather, he seems to be posting his answers to questions where they are relevant. (In particular, his repost in the cosine question addresses the part of the original question asking about general methods.)
What's shameful about having an minor over-reaction by a moderator corrected?
A common and difficult moderating situation is when a user does a series of several problematic things and the moderator reacts to the whole situation, while people coming along only see part of it. Annix has a history of making genuinely false statements (this is a separate issue than using and clearly stating when certain steps are difficult to justify rigorously), which is why his posts attracted reasonable moderator attention.
It is crucial to having a functioning website that we keep our signal to noise ratio high. If we needed to wade through 20 false proofs of the twin prime conjecture every day MO would be useless.
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