Dave

Well, I suppose we just don't want to bother writing down loads of compositions, so we come up with a shorthand way of doing it. Multiplication just isn't as useful as composition

Yes, but you can simplify that quite a bit. Notice that by factoring the -1 out, you get the much nicer [imath]g^{(n)}(x) = (-1)^{n+1} ( ne^{-x} - xe^{-x})[/imath].

Generally, we interpret gn as: [math]\underbrace{g \circ \cdots \circ g}_{\text{n times}}[/math] The circle indicates composition of the function. This is mainly because mathematicians are lazy, and in general we're more interested in composition than multiplication. We use [imath]g^{(n)}[/imath] to denote the n'th derivative. Pretty much

Google is your friend. Just note that arcsin = sin-1, arccos = cos-1 etc.

That's the idea. It's blatently easy because you can just differentiate from the statement of n=k being true.

Note that [imath](-1)^{n+1}[/imath] is -1 when n is even, and 1 when n is odd. It's a common thing to find. Obviously if you haven't covered stuff like the chain rule and all that, you don't need to go on and prove it by induction. I had assumed that you were at some kind of 1st year university level maths or something

Also depends on the content. But yes, you'll probably have to get someone to pay to read it.

Another update: Looks like most of the stuff we need to make this work is already there. I just need to sort out styles and the LaTeX mod. I'll talk to blike about it when I see him next.

I don't think this is a good form for the answer. You want an explicit formula in terms of x, and whilst you have one there, it's much neater to write down something like [imath]g^{(n)}(x) = nx + b[/imath] or whatever.

That's not entirely correct. For example: [imath]g'(x) = e^{-x} - xe^{x}[/imath] [imath]g''(x) = -2e^{-x} +xe^{x}[/imath] You must make sure to observe your negative signs Also, you might well see a pattern, but that certainly isn't a proof. Whilst it's an easy inductive proof, it is required to make your observation concrete.

What it's looking for is an explicit formula (in terms of n and x) for the n'th derivative of g. How do you approach it? The way I would consider is to work out the first few terms. Also, consider writing g as [imath]g(x) = xe^{-x}[/imath]. Once you've worked out the first few terms and figured out what the pattern is, make a guess at what you think the formula is. You can then prove it using induction (which is an easy proof).

I find that whenever reading mathematical texts (or pretty much any scientific text) it's always helpful to look out for anything that says 'clearly'. Most of the time, it's obvious. But whenever it isn't, it's always a good idea to go through that step and expand it out by yourself to make sure that you understand every single thing that's going on. When you don't do this, your understanding tends to drop quite a lot. In maths at least, it's one thing to read and understand a text, but quite another to actually use the content to prove things and do useful things with it.

I would tend to agree on that one. The effects of temperature would be completely negligable as the remaining air was blown out into space.

Moved as requested.

I've found that I've had to hard-reset these things quite a lot as well. They can be a little annoying at times

I get the impression that blike will be setting up a Wiki, but he's really busy at the moment with med school and the like. I like your ideas of welcome messages and all that, and I'll certainly look into this as a "quick fix" for the time being. I'll also look into having a little bit more excitement on the front page, because I do agree that it's quite dull at the moment. We need some pretty pictures or something on there. Besides all of this, I feel that we need to concentrate on getting more users on here. So this is a call to all of our users really: try and promote SFN! Obvously, we don't want you to spam it about everywhere, but if you're chatting with a friend or whatever, then mention it. I'd like to think that this community is one of the best out there, so whatever you can do to promote SFN would be great. Especially, in my case, the maths section

There's a couple of useful changes for the mods (inline moderation, some assorted AJAX stuff etc), but mostly the work is behind the scenes. I don't think we're going to be upgrading immediately, but we'll certainly upgrade once all of the mods we use are ported over to the new plugin system.

We've certainly discussed this idea before, but as of yet not implemented it. We get a little bit of money from the Google ads and the advertisements at the bottom of the page - certainly enough to cover hosting for the time being. The idea was to have this extra subscription to show people who actively support SFN with a 1-time payment, but I think for the time being it's been shelved. A major reason, for me at least, is that people who sign up for this may feel that they are superior to others, and this will only lead to divide the community. Also, people who have paid for subscription and then have their membership suspended may cause us quite a significant pain in the ass. It's something we'd have to consider, because it's quite likely to happen at some point. I'll comment on the suggestions though, just for clarity's sake. 1. I'm not sure the mods would approve of this one. People who pay could theoretically close their threads if they were losing an argument. I, for one, don't like this. It's the equivalent of saying: "You're wrong, I'm right, end of discussion." Obviously they could be re-opened, but that's not the point. 2. That's fine. I think we would probably re-enable editing of posts also. 3. Fine again. 4. Don't think we'd allow this. Makes the threads look disjointed, also there is a distinct possibility for abuse. Mainly aesthetical though. 5. We were considering a different coloured username or something to that extent. Something that looks cool anyway 6. Perhaps. Depends on how it affects the way the posts look. 7. Obviously we'd allow this. 8. Not pre-cognitive, I'm afraid Hope you find this useful. I'm not trying to shoot your ideas out of the water, but I think at least for the next couple of months we won't have this. Have to speak to blike to confirm this though.

I wouldn't say it's a fundamental part of learning complex numbers (I didn't learn about it until the very end of my Further Maths A-level) but it's a nice trick to know. Another way of doing it is to say that [imath]\sqrt{a+bi} \equiv c+di[/imath] for some [imath]c, d \in \C[/imath] and then square both sides, equate co-efficients. That way is a little more tricky (you run into some problems with positive/negative square roots) but works equally as well.

That's the idea. I think the argument I saw a long time ago said that basically no matter which k we pick, we can always find x or y big enough so that 1/|xy| > k, so there could never exist a k big enough.

If you click on the "SFN Chatroom" link at the top of the page, then this should send you to the chatroom in question

It certainly looks to be the correct approach.

There's something in the order of 6,000 posts that are in the Removed Thread area. This is where most of the stuff that we delete goes.

Easiest way to do it is by looking at the polar form. [imath]i = \cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2})[/imath]. Now square root: [imath]\sqrt{i} = \left( \cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}) \right)^{\frac{1}{2}}[/imath] Apply de Moivre: [imath]\sqrt{i} = \cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4}) = \frac{1+i}{\sqrt{2}}[/imath].

I know it's a bit late, but I thought I should probably mention something about this at least. Also, I'm pretty useless at this sort of thing, but I thought I should give it ago. As you probably know, we passed the 200,000 post mark about 2 weeks ago. Personally, I think it's a magnificent milestone for SFN, but I'll leave the talking to blike when he gets around to posting here. My thanks to everybody that's contributed to SFN, taken time to post and contribute to the community. This place wouldn't be the same without it, and the more people we get here, the better it will get. I don't know any other place that has such a great atmosphere in helping others and discussing the latest issues in all the various guises of science. Also, a big cheer for the moderator staff is in order, since this place wouldn't be half as good as it is today without them. I don't know really what else to say about them without sounding very weedy and silly, but they do a great job. It takes a great deal of time and is often very frustrating (from personal experience), but their contributions are still magnificent as usual. Also, a quick aside to congratulate Pangloss and Mokele on their promotions, which I believe were much deserved. So, that's about it from me. Here's to the next 100,000 milestone!