Wikipedia:Reference desk/Archives/Mathematics/2007 April 5

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April 5

square root of pi

My siblings sometimes mention the square root of pi to make fun of me when I get too technical in my explanation of something. I was wondering if the square root of pi actuall has any use anywhere in mathematics.J.delanoy 01:31, 5 April 2007 (UTC)

One in in the probability density function of the normal distribution. (Note you can factor out the root of 2.) There are many others if you look around... Baccyak4H (Yak!) 01:55, 5 April 2007 (UTC)

Error_function, although this is the reciprocal of ${\sqrt {\pi }}$--Ķĩřβȳ♥♥♥ŤįɱéØ 02:45, 5 April 2007 (UTC)

"square root of pi" gives 15100 Google hits. I have no idea how many have use in mathematics. PrimeHunter 02:55, 5 April 2007 (UTC)

Γ(1/2), and Stirling's approximation. —David Eppstein 03:14, 5 April 2007 (UTC)

It shows up when you try to square the circle. It's the width of a square with the same area as a circle of radius 1. In fact, that's why squaring a circle is impossible using ruler and compass - Pi is transcendental. Black Carrot 04:03, 5 April 2007 (UTC)

I don't know much about ${\sqrt {\pi }}$, but apparently $\left({\sqrt {\pi }}\right)^{2}$ has a lot of use. --Hirak 99 15:16, 5 April 2007 (UTC)

That's hilarious, NOT.J.delanoy 23:47, 5 April 2007 (UTC)

$\int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}$ Gaussian integral --Spoon! 19:32, 5 April 2007 (UTC)

Thanks for answering the question!!!! You guys ROCK!!!!!J.delanoy 23:49, 5 April 2007 (UTC)

And gamma(1/2) = (-1/2)! = root pi→81.153.220.170 11:52, 6 April 2007 (UTC)

Triangle integrals?

The Riemann, Darboux, Lebesgue, etc. integrals use rectangles. By are there any integral definitions which use triangles? After all, triangles are very "special" and have their own study (trigonometry). Couldn't the area under a curve be calculated using triangle approximations? All polygons can be broken up into triangles, but not all polygons can be broken up into rectangles. So wouldn't triangle integration converge towards the actual value much quicker than Riemann integration? Thanks.--Ķĩřβȳ♥♥♥ŤįɱéØ 04:28, 5 April 2007 (UTC)

Sure. Take any rectangular integral, and divide each term by two. Then multiply by two. This corresponds to slicing each rectangle from one corner to another, and reassembling the triangles. Or, measure it with trapezoids (which are triangle-like) or, for something even more fun, try wiggly segments of polynomials. There are loads of choices, and I wouldn't be surprised if it was possible to define integrals using pretty much any shape you want. For instance, applying your idea about polygons, it's possible to find the area of a circle by essentially integrating over triangles. Probably other blobby shapes too. It wouldn't be all that different from polar integrals, which use wedges of circles. Then there's integrating solids of revolution, which use washers and disks. The big question in each case is, is this really the easiest way to do this? Rectangles are useful because, in a rectangular coordinate system, they're really really easy to keep track of. If you'll think back to trigonometry, you may recall that powerful as the system is, it's anything but convenient. Black Carrot 05:59, 5 April 2007 (UTC)

So is it because of the orthogonal coordinate system that makes rectangles simpler? If, say, we wanted to find integrals in non-orthogonal coordinate systems, would triangles be better(Like, if instead of the x and y axes being perpendicular, if they made a 45° angle)?--Ķĩřβȳ♥♥♥ŤįɱéØ 06:12, 5 April 2007 (UTC)

That's right. For example in polar coordinate system the area would be easily integrated by triangles, which approximate sectors between some points $(r_{i},\theta _{i})$ and $(r_{i+1},\theta _{i+1})$ with common third vertex at the system's origin $(0,0)$. --CiaPan 06:19, 5 April 2007 (UTC)

You can dissect a rectangle into a trapezoid and a triangle. Remove the triangle and you get a trapezoidal approximation. For an example see Riemann sum#Trapezoidal rule. This in fact is almost (i.e. for partitioning into small sub-intervals) equivalent to Riemann sum#Middle sum. For some cases the Simpson's approximation with parabolas is even better than trapezium method. --CiaPan 06:15, 5 April 2007 (UTC)

Thanks for all the information. I thought up triangle integrals in the shower today... yes, I am that "weird". In any case:

This picture answers my question.--Ķĩřβȳ♥♥♥ŤįɱéØ 06:31, 5 April 2007 (UTC)

Now keep in mind, those aren't triangles in the picture, they're wedges from circles. It's just that triangles give almost the same answer if they're narrow and exactly the same answer in the limit. Black Carrot 07:59, 5 April 2007 (UTC)

yes, those are sectors, but their similarity with a triangle shape makes triangles a possible surrogate.--Ķĩřβȳ♥♥♥ŤįɱéØ 09:26, 5 April 2007 (UTC)

how to undestand this?

suppose we want to devide($17)up on 3 shares,the first share is(1/9),second share is (1/3)and the third share is(1/2).abviously we cannot get integer shares.NOW,add (1+17=18),the three shares of(18)would be(2,6,9)respectively.we can see(9+6+2=17)which means that we get fully three shares without losing the additional($1).how can we understand this in mathemetical way?80.255.40.168 11:59, 5 April 2007 (UTC)ARTHER

Suppose you were asked to split $100, giving 1/3 to one person and 1/3 to another person. You could borrow $50 so that you have $150, then give 1/3 of this to each person ($50 each), and still have $50 left to repay the loan. Of course, you haven't really split the $100 into 1/3 and 1/3, because 1/3+1/3 is not equal to 1 - it is only 2/3. Your problem is just a slightly more complicated version of this $100 split. Notice that 1/9 + 1/3 + 1/2 is not equal to 1 - it is only 17/18. Gandalf61 13:31, 5 April 2007 (UTC)

On the other hand, in the example above, you are not really splitting $100 into thirds, you are splitting $150.J.delanoy 23:51, 5 April 2007 (UTC)

Perhaps David Eppstein can clarify the history of this classic use of Egyptian fractions, but some version of this puzzle has been challenging and entertaining minds for a very long time. I believe I first saw it as a tale of Nasrudin, but it was already old by then. --KSmrq^T 20:10, 5 April 2007 (UTC)

Initial Value Problems convergence etc

hi, i'm going through my lecture notes (on Numberical solutions of ODEs btw) and i cant get what my lecturer has done. I wont write out the whole thing becuase a) its 3 pages long and b) i dont have latex. but i'm guessing what i'm stuck on isn't IVP specific but maybe it is, i dont know... anyway, heres an outline... we're dealing with Eulers method which we've found the local truncation error for, we're also told that f(t,y) satisfies a lipschitz condition so

|[f((tj), y(tj)) - f((tj), zj)] | ≤ L |y((tj) - (zj) |.

So we keep chunking through, doing our thing until we get to:

|ej+1|≤ (1+Lh)|ej| +½h2k

so we're then supposed to replace j with j-1 sp predictably enough we get:

|ej|≤ (1+Lh)|ej-1| +½h2k  (*)

she then says that (*) is less than

(1+Lh){(1+Lh)|ej-2|+½h2k} +½h2k

wtf?

If anyone knows whats going on i'd be so appreciative. (Apologies if i'm not being very clear or its not readable). thanks 130.88.52.26 12:32, 5 April 2007 (UTC)

That's the result of writing your (*) expression again — expressing $|e_{j-1}|$ in terms of $|e_{j-2}|$ — and substituting into (*). --Tardis 14:55, 5 April 2007 (UTC)

ok but where does the extra 1+lh come from?130.88.52.11 17:04, 10 April 2007 (UTC)

The same place the extra $h^{2}k \over 2$ comes from? You're nesting two "calls" to (*). --Tardis 21:20, 10 April 2007 (UTC)

Famous function?

Does the graph of $x/(1+x^{2})$ have any specific name or function?Wbchilds 13:25, 5 April 2007 (UTC)

It's related to the Lorentzian function; in fact, integrating it over all reals would yield the mean of that probability distribution if it existed. --Tardis 15:15, 5 April 2007 (UTC)

Volume of sphere

How to calculate the volume of a 'perfect' sphere besides putting it into a jar and see the rise in the water volume? If so, what is the formula? —The preceding unsigned comment was added by Invisiblebug590 (talk • contribs) 13:51, 5 April 2007 (UTC).

${\frac {4}{3}}\pi r^{3}$ where r is the radius of the sphere. See sphere. x42bn6 Talk 14:02, 5 April 2007 (UTC)

And where π is tasty. − Twas Now ( talk • contribs • e-mail ) 17:21, 5 April 2007 (UTC)

Proof that Ke^x for contants K are the only functions that are their own derivative

The article exponential function correctly says that functions of the form $f(x)=Ke^{x}$ for contant K are the only functions that are their own derivative. Any suggestions on a link to a proof of that uniqueness? Dugwiki 16:11, 5 April 2007 (UTC)

Assume that f is a differentiable function with f(0) = 1, and put g(x) = log f(x). Then by the chain rule g'(x) = f'(x)/f(x), which equals 1 if (and only if) the functions f and f' are the same. Function g'(x) is then Lipschitz continuous. Now apply the Picard-Lindelöf theorem to obtain g(x) = x, and use f(x) = exp g(x). The extension to other values of K is not hard. You don't actually need this sledgehammer to swat this gnat; all you need is that only constant functions have a derivative that is 0 everywhere.  --Lambiam^Talk 16:28, 5 April 2007 (UTC)

In fact, as I mentioned below, the solution using the Picard-Lindelöf theorem appears to be even simpler. Set $f(t,y(t))=y(t)$ (the identity function) and y(0)=K and y'(t)=f(t,y(t))=y(t). Then PLT says that $y(t)=Ke^{t}$ is the unique solution. No need to even mess with the chain rule there. Dugwiki 17:52, 5 April 2007 (UTC)

It's the only solution of the differential equation $dy/dx=y$. Basically you get $dy/y=dx$, and integrating $\ln(y)=x+C$. From there your equation follows. For more detailed proof, see Lambiam's links above. --Hirak 99 16:35, 5 April 2007 (UTC)

PS. Unless you allow complex values for the constant of integration $C$, you won't explain why negative (and for that matter, complex) $K$ works in your equation. --Hirak 99 16:40, 5 April 2007 (UTC)

Yeah, someone else pointed out my error on the article discussion page. Basically the PLT says that $y(t)=Ke^{t}$ is the unique solution to the differential equation $y'(t)=y(t),y(0)=K$ with $f(t,y(t))=y(t)$. Dugwiki 17:41, 5 April 2007 (UTC)

P.S. I almost overlooked Hirak's comment about negative K. I think you are reading what I typed as "$(Ke)^{x}$" (which doesn't make sense in the reals if K is negative). That's not what I mean, though. I'm talking about "$K(e^{x})$", which is well defined for all real K. Dugwiki 20:05, 5 April 2007 (UTC)

I think Hirak referred to the relationship C = ln(K), which, if C is real, implies K > 0. To avoid the issue I assumed f(0) = 1 in my first reply.  --Lambiam^Talk 21:01, 5 April 2007 (UTC)

Ok, I thought he was replying to my post. Thanks for clearing that up. Dugwiki 21:06, 5 April 2007 (UTC)

Actually, $dy/y=dx$ integrates to $\ln |y|=x+C$ ($\ln |y|$ is a better antiderivative of $1/y$ because it works for negative numbers). So the absolute value around $y$ ensures that for every solution $y=f(x)$, $y=-f(x)$ is also a solution. Which accounts for the negative K's.

Also, when you divided by $y$ initially (like when you divide by any expression), you need to separately consider what happens when the thing you are dividing by is zero. When $y=0$, that is indeed a (trivial) solution to the equation, so that accounts for K=0. --Spoon! 10:48, 6 April 2007 (UTC)

Which Statistical Test?

I've done a test with 3 groups, one control (in this case water) and two experimental (two different kinds of drinks) and measured final time for a distance covered. I know how to standard deviations and all that but after that i am slightly confused, what do i have to do after this to find any statistical difference between all the groups, will i need a t-test, ANOVA or some other test? If it is an ANOVA could someone explain how to do it in a laymans way or put a link up to somewhere that does, i can never make sense of the mathmatical stuff on wikipedia.

Any help would be appreicated. Rickystrapp 19:45, 5 April 2007 (UTC)

I won't comment on the math involved, but will comment on your control. The subjects presumably can tell if they are drinking water or one of the drinks. This makes the control not very valuable, as the placebo effect can come into play. Ideally, the study should be double-blind, where neither the experimenter nor subjects can identify whether they have been given the control until the results are in. In the case of water, at very least you could add dye to make it look like it might be something else. If you can find an additive you're sure will change the taste and smell without altering performance (perhaps you know this from a previous study), you could also add that to the control. StuRat 20:05, 5 April 2007 (UTC)

I did take that into consideration when designing the study, however i decided it didnt matter enough, i wont go into my experiment but it was agreed not to do that but mention it in the limitations of the study. Rickystrapp 20:26, 5 April 2007 (UTC)

If the only variable in the experimental set-up under your control was the type of potion quaffed, then I see no use for analysis of variance. If the data in each group looks like it follows a normal distribution – you can use Shapiro-Wilk or Kolmogorov-Smirnov to test for that – then indeed Student's t-test is a reasonable test for significant differences. In view of the origin of the test it would be quite fitting if one of the experimental elixirs was Guinness. You should decide on the confidence interval before you compute the test statistic.  --Lambiam^Talk 21:16, 5 April 2007 (UTC)