Wikipedia:Reference desk/Archives/Mathematics/2010 September 24

= September 24 =

Limit
Hello. If $$a > 0$$, why does $$\lim_{x\rightarrow 0^{+}} x^a \ln x = 0$$? Thanks in advance. --Mayfare (talk) 03:01, 24 September 2010 (UTC)
 * One way to show it is to write $$x^a\ln x = \frac{\ln x}{x^{-a}}$$ and use L'Hôpital's rule. -- Meni Rosenfeld (talk) 03:35, 24 September 2010 (UTC)
 * You can use a change of variables y = x−a:
 * $$\lim_{x\to0+}x^a\ln x=\lim_{y\to+\infty}\frac{\ln y^{-1/a}}y=-\frac1a\lim_{y\to+\infty}\frac{\ln y}y=0.$$
 * —Emil J. 13:18, 24 September 2010 (UTC)

Using 2d complex vectors to represent 4d space
I'm trying to represent 4d real space with vectors like


 * $$X=\begin{bmatrix} w + ix \\ y + iz \end{bmatrix}$$

and am furthermore trying to define something analogous to a rotation ($$O(4,\mathbb{R})$$) matrix for such vectors.

Any analogue of $$O(4,\mathbb{R})$$ must preserve length. Using the hermitian form


 * $$(X,X)=X^\dagger X$$

as a measure of length, one can deduce that $$(X,X)$$ must equal


 * $$(AX,AX)=X^\dagger A^\dagger AX$$

For equality, it seems plain that $$A$$ must be a unitary matrix. But this clearly doesn't represent to full group of rotations, as $$O(4)$$ is 6-dimensional, whilst $$U(2)$$ is only 4-dimensional. Where am I going wrong?--Leon (talk) 08:01, 24 September 2010 (UTC)
 * The condition $$A^\dagger A=1$$ is sufficient for $$X^\dagger X=X^\dagger A^\dagger AX$$, but is it really necessary? Bo Jacoby (talk) 09:02, 24 September 2010 (UTC).
 * I'm not sure, but I do need to preserve lengths to represent rotations. Is there a more general group that satisfies that condition?--Leon (talk) 09:10, 24 September 2010 (UTC)
 * On reading the article "General linear group", it appears that $$U(2)$$ is the maximal compact subgroup of $$GL(2,\mathbb{C})$$, and given that $$O(4,\mathbb{R})$$ is compact it would appear that I can't get anything quite like a rotation representation. Is there some non-compact covering group of the rotation in group that I can use in this fashion?  Thanks.--Leon (talk) 10:44, 24 September 2010 (UTC)

Is there a way to analytically find the inverse of y=x^x on [1,∞)?
I've been on this for a while and haven't gotten anywhere. I've tried starting with y=x^x, swapping x and y to get x = y^y, and trying to get an equation for y in terms of x. But I can't get any equation for y that doesn't have y in it. Thanks. 20.137.18.50 (talk) 13:33, 24 September 2010 (UTC)
 * You can do it with the Lambert W function, which is not elementary: $$y=\frac{\ln x}{W(\ln x)}$$. -- Meni Rosenfeld (talk) 13:40, 24 September 2010 (UTC)
 * (e/c) The inverse is y = exp(W(log x)), where W is the Lambert W function. You cannot write it in terms of more elementary functions.—Emil J. 13:42, 24 September 2010 (UTC)
 * Thank you both. It's a relief to me that the answer is something I never would have known and not that I missed something simple. 20.137.18.50 (talk) 13:48, 24 September 2010 (UTC)

Stirling's approximation
Is the Stirling approximation of a noninteger n meaningful, that is, does the factorial of a noninteger n exist? 24.92.78.167 (talk) 21:29, 24 September 2010 (UTC)
 * The natural extension of the factorial function to nonintegers is the gamma function. Algebraist 21:32, 24 September 2010 (UTC)
 * ..and of course the analogous Stirling's approximation for complex numbers holds true. (PS: 24.92: it's not a criticism, but had you put your very title in the wiki's search engine, and read the article, you'd had found everything you needed in few nanoseconds) --pm a 09:48, 25 September 2010 (UTC)

Learn Calculus
Hi, it's me again. I've done some independent calculus study before and I know how to find limits and integrate and differentiate and things in that nature. I'm planning to take a calculus exam next year for placement into a class. The exam will cover the equivalent of a first year and anywhere between 1/2 to 3/4 of second year calculus course. I need to study for this independently/alone in addition to work from my other classes. Does anyone recommend any books that I can use to fill in the foggy spots and extend what I know? These shouldn't require any more than a strong grasp of the prerequisites to calculus. Thanks. 24.92.78.167 (talk) 21:58, 24 September 2010 (UTC)
 * If you're looking for a textbook then Calculus by James Stewart is what we used in 1st and 2nd year (I'm an engineer). I don't know what it's like to go through completely independently without having attended any lectures first, when I open it now for some light reading and derivation I find the explanations very easy to follow. He tends to use lots of visuals and examples in explaining concepts, which gives you the intuitive grasp and understanding quite quickly. It's perfect if you're an engineer or physicist, but may not be rigorous enough if you're a mathematician. (The real analysis is tucked away in the Appendices and not all of it is covered.) Zunaid 22:18, 24 September 2010 (UTC)
 * I happen to own Stewart (2nd ed.) and agree it's as good as any. The market for calculus textbooks is highly competitive and publishers do their best to put out quality, if somewhat homogeneous work. So you can't go too far wrong by taking pot luck at a used book store. If you're teaching yourself then you might want to get a few books, since you'll understand the concepts better if you see them explained in more than one way.--RDBury (talk) 23:27, 24 September 2010 (UTC)

I forgot to mention this should not be like a textbook, or something very expensive. I'm just looking for a guide, and I'm not a mathematician (or even an engineer) I am a student. 24.92.78.167 (talk) 00:15, 25 September 2010 (UTC)
 * Maybe a Schaum's Outline is what you're looking for? Some of your comments make it sound as if your purpose is to pass some stadardized test.  (Those are instruments of Satan, by the way.  But never mind that for now....)  There are many calculus books that are nearly identical; I don't know why people write those.  The ones that are unusual---those I've seen---are mostly for people with an intellectual interest in the topic rather than to pass a standardized test (Spivak, Apostol, one other whose author's name escapes my just now).  Maybe an exception to that is Jerome Keisler's book Elementary Calculus, which you can download off the web.  Its unusual nature is that it relies explicitly on infinitesimals.  I think those help intuition. Michael Hardy (talk) 21:28, 25 September 2010 (UTC)
 * Oh: There's also Sylvanus Thompson's book Calculus Made Easy, which is mainly for those whose interest is in using calculus in engineering and the sciences. Michael Hardy (talk) 21:30, 25 September 2010 (UTC)
 * I find this conjunction a bit odd. The primary target audience of textbooks is students. -- Meni Rosenfeld (talk) 17:25, 25 September 2010 (UTC)

http://www.math.rutgers.edu/~zeilberg/DrZhandouts.html Count Iblis (talk) 02:35, 25 September 2010 (UTC)


 * There are lots of inexpensive calculus study guides published. One series is Schaum's Outlines. I would estimate you could pick up something like this for maybe $20 or $25, rather than a full-on calculus textbook which could cost $150 or more. If you have access to a university bookstore you should be able to find a selection of these study guides there. —Bkell (talk) 03:07, 25 September 2010 (UTC)


 * I don't know where you live but if you have any bookstores like "Half Price Books" by you, those would be great. I found an older edition of Stewart's Calculus  (4th edition) for $7 and that was maybe 3 or 4 years ago.  You don't have to get that one, as pointed out above, they are very similar usually but such a store would be likely to have some sort of textbook at some point if you check back every once in a while.  Also, there are lots of things you can find online.  Here are some online notes on Algebra, Calc 1-3, Linear Algebra, and Differential Equations [].  I just searched for "calculus online" (where I didn't use the quotes) on Google and that's one of the first things to come up. StatisticsMan (talk) 02:20, 27 September 2010 (UTC)


 * Don't forget wikibooks:Calculus of course. 67.119.2.101 (talk) 06:43, 27 September 2010 (UTC)