Wikipedia:Reference desk/Archives/Mathematics/2010 March 29

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March 29

Rouché's theorem and real solutions

Hi all,

I'm trying to use Rouché's theorem to show that all the solutions to $z\sin(z)=1$ are real, and I've shown (fairly trivially) that in the interval $[-(n+1/2)\pi ,\,(n+1/2)\pi ]$ there are 2(n+1) real roots, and now want to use Rouché to show that in the disc $D(0,(n+1/2)\pi )$ there are also 2(n+1) roots, so they must -all- be real. I'm using the 'symmetric form' of Rouché's theorem as referred to in the article, and trying to show that $|(z\sin(z)-1)-z\sin(z)|=|g-f|=1<|z\sin(z)|=|f|$ on the boundary of the disc, $\partial {D}=\{z:|z|=(n+1/2)\pi \}$, which would imply f ; but I'm not sure how to actually go about showing this (if indeed it is correct). It seems like it should be correct, because that would give me the correct number of zeroes in the disc (counted with multiplicity - i.e. 2 at z=0); 2(n+1) roots of zsin(z)=0 in the disc - but I can't see how strictly to prove it. I worked out that $|\sin(z)|={\sqrt {\sin ^{2}(x)+\sinh ^{2}(y)}}$ where z=x+yi, and I can see graphically the minimum for this is roughly around $y=0,\,x=\pm (n+1/2)\pi$, but I can't see how to prove it, and of course I'd rather not get into lots of messy calculus of minima if it can be avoided. Could anyone suggest anything? :)

Thankyou very much, it's greatly appreciated! Spalton232 (talk) 15:37, 29 March 2010 (UTC)

Hints. Consider the strip {z : |Im(z)|<1}. Prove that there are no zeros of g(z):=zsin(z)-1 outside of it (use the expression for |sin(z)| you wrote, and note that |sinh(y)|>1 for any real y with |y|≥1). Take f(z):=z sin(z). Compute all zeros of f, with multeplicity. Prove that |f(z)-g(z)|  = 1  <  |f(z)| holds true either if |Im(z)|=1 or if |Re(z)|=(k+1/2)π (for any positive integer k), and use Rouché's theorem in consequence. Conclude as you were doing, counting with multeplicity the real zeros of f and g in the interval |x|≤(k+1/2)π. --pma 08:04, 30 March 2010 (UTC)

Sorry, perhaps I'm being slow; I follow that, but I don't see how that helps with the region of |Im(z)|<1, except on the x-axis; I understand why |Im(z)|>=1 satisfies 1<|zsin(z)|, and why on the real axis it holds, but what about the region 0<|Im(z)|<1? Maybe I'm missing something obvious, but I still can't seem to rule out that region with any obvious inequalities. Sorry I don't seem to be getting this, hopefully you're still checking this far back on the reference desk! Thanks very very much - Spalton232 (talk) 22:20, 1 April 2010 (UTC)

Yes, it seems you just missed the last small step. Let k be a positive integer, and consider the rectangle R:={z : |Im(z)|≤1 and |Re(z)|≤(k+1/2)π}. Rouché's theorem applies to f(z) and g(z) on R (you have to check it), and tells you that they have the same number of zeros in R. You know by direct computation that there are exactly 2(k+1) zeros of f(z) in the rectangle (i.e. integer multiples of π, all simple but 0, that has multiplicity 2). Therefore, g(z) also has exactly 2(k+1) zeros in R. But you have already found 2(k+1) zeros of g(z) in the interval [-(k+1/2)π, (k+1/2)π], hence there is no other zero of g(z) in R. Since k is arbitrary, you conclude that all zeros of g(z) are real. Is it ok? (PS: in our link, Rouché's theorem is quoted with a bounded domain K with a smooth boundary, but in fact no smoothness is needed, and a rectangle works perfectly). --pma 20:15, 3 April 2010 (UTC)

Birthday Probabilities

I have some probability questions. One leads on from the other.

1. What is the probability that in a room of n people, exactly two people will share a birthday?
2. What is the probability that in a room of n people, at least two people will share a birthday?
3. What is the probability that in a room of n people, k people will have a birthday falling anywhere within a period of p consecutive days?

In regards to the last point, for example, what is the probability that in a room of ten people, six of them have birthdays falling in a given week? Although I would like a general solution; if at all possible. Thanks in advance. •• Fly by Night (talk) 19:40, 29 March 2010 (UTC)

Birthday problem may help 129.67.37.143 (talk) 20:21, 29 March 2010 (UTC)

It helps with the first question. I am primarily interested, as my example might show, in the third question. I hoped that a logical sequence of questions might help the thread. I know that by the strictest sense of the reference desk that I ought to be pointed to article so that I might help myself; but I was hoping for a more contemporary style reference desk answer, i.e. a solution with some links to help me understand the solution. Don't worry: it's not a homework question. I'm far too old for homework. •• Fly by Night (talk) 21:40, 29 March 2010 (UTC)

Actually it deals with all of your questions, but only the k=2 case of q3 - have a look at "near matches". There's a reference in that section that might be helpful with the arbitrary k case, and another to an article called "generalised birthday problem" that might help. I'm sorry I can't give you a better solution. 129.67.37.143 (talk) 23:19, 29 March 2010 (UTC)

Thanks for making an effort. But it doesn't actually deal with all of my questions. It deals with a very special case of my third question. Special cases are not very useful really. The (positive) number 2 is both even and prime. How many other (positive) prime numbers are even? If anyone can answer my questions as written, i.e. give a proper solution to the questions, then I would be very grateful. •• Fly by Night (talk) 23:32, 29 March 2010 (UTC)

If you assume that ever day of the year has the same probability of birth, which is an acceptable assumption, then there is a 1/365.25 chance that a person is born on some given day. If you are looking at a 2-day span, there is a 2/365.25 chance that a person is born on those two days. If you look at a 3-day span, there is a 3/265.25 chance. If you look at p days, there is a p/365.25 chance. If you have 1 person with a p-span of days, there is a p/365.25 chance that the person will be born on that day. If you have 2 people, there is a p/365.25 chance per person, for (p/365.25)². If you have 3 people, there is a (p/365.25)³ chance of the three being born on those p days. For k people, there is a (p/365.25)^k chance. Is that what you are asking? Obviously, this would be much easier without leap years. -- kainaw™ 23:41, 29 March 2010 (UTC)

This is almost there. I want to start with n people and I want exactly k of them to have a birthday within a fixed period, of say p days. This is different to having n people and wanting them all to have a birthday within the same period of p days. The must be some kind of averaging or something like that... •• Fly by Night (talk) 00:10, 30 March 2010 (UTC)

Is this not given by the multinomial distribution? 66.127.52.47 (talk) 02:55, 30 March 2010 (UTC)

If you fix a particular week, then the probability that k people from n have birthdays in that week is multinomial. However this can't be used immediately to solve qn3 because the probability that k people have a birthday in a period w_1 and the probability that k people have a birthday in period w_2 are not independent. Inclusion-exclusion is needed. Let W_p be the set of all periods of length p consecutive days, and if w is such a period B_w be the event that at least k people have their birthday in w. Qn 3 asks for $P(\bigcup _{w\in W}B_{w})$. Inclusion-exclusion (and some kind of argument inductive on p) can be used, but it won't be straightforward. 129.67.37.143 (talk) 09:49, 30 March 2010 (UTC)

Ah, ok. Multivariate hypergeometric distribution then? 66.127.52.47 (talk) 20:55, 30 March 2010 (UTC)