Wikipedia:Reference desk/Archives/Mathematics/2011 November 6

= November 6 =

Finding associated Legendre polynomials
The method of finding associated Legendre polynomials by differentiating Legendre polynomials is given in this book on filters as;


 * $$P_{N,m}(\omega) = \frac {d^m}{d\omega^m} P_{N+m}(\omega).$$

This does not seem to agree with the Wikipedia article. Can anyone explain this discrepancy?  Spinning Spark  11:40, 6 November 2011 (UTC)

I understand this to mean,


 * $$P_N^{(m)}(x) = \frac {d^m}{dx^m} P_{N+m}^{(0)}(x)$$


 * where $$P_N^{(m)}(x)$$ is the m-th associated Legendre polynomial of Legendre polynomial order N.

Is this really irreconcilable with the expression in our article or have I made a mistake somewhere?  Spinning Spark  11:48, 12 November 2011 (UTC)

Applying the chain rule more than once
I am confused about how to apply the chain rule to a composite function with more than 1 nested function. How would I apply it to this example: $$f(x)=g(h(i(x)))$$? I know that $$f'(x)=g'(h(i(x))) \times h'(i(x))$$. --Melab±1 &#9742; 20:33, 6 November 2011 (UTC)
 * What you wrote isn't quite right. You should say: $$f'(x) = g'(h(i(x))) \cdot \frac{d}{dx} h(i(x))$$. Now you find the latter derivative there by the usual chain rule. Staecker (talk) 20:42, 6 November 2011 (UTC)


 * Just apply the rule step-by-step:
 * $$ \left[ f(g(h(x))) \right]' = \left[g(h(x))\right]'\cdot f'(g(h(x))) = h'(x) \cdot g'(h(x)) \cdot f'(g(h(x))) \, . $$ — Fly by Night  ( talk )  21:03, 6 November 2011 (UTC)
 * So if $$f(x)=(cos(3 \times x))^2$$ then $$f'(x)=[2 \times cos(3 \times x)] \times [-sin(3 \times x)] \times [3]$$? --Melab±1 &#9742; 21:14, 6 November 2011 (UTC)
 * Yes: $$ \frac{\operatorname{d}}{d\! x} \cos^2(3x) = -6\sin(3x)\cos(3x) \ldots (= -3\sin(6x)) \, . $$ — Fly by Night  ( talk )  21:37, 6 November 2011 (UTC)

perfect number
What is the application of this numbers? — Preceding unsigned comment added by 203.112.82.128 (talk) 20:58, 6 November 2011 (UTC)
 * None. Give him threepence since he must profit from what he learns ;-) Dmcq (talk) 21:29, 6 November 2011 (UTC)
 * Then why bother finding this numbers? — Preceding unsigned comment added by 203.112.82.1 (talk) 22:15, 6 November 2011 (UTC)
 * What's the application of poetry? 109.150.108.80 (talk) 23:23, 7 November 2011 (UTC)
 * I think the interest might be generated by the claim of "unsolved problems", or the reward of $150,000 for the related Mersenne prime (see Great Internet Mersenne Prime Search). Large prime numbers (not necessarily Mersenne) have found a very wide application in computer security (see RSA and Public-key cryptography).   D b f i r s   00:25, 7 November 2011 (UTC)

Binomial related function?
What is the closed form for this function? k fixed. --HappyCamper 21:06, 6 November 2011 (UTC)

$$f_k(q) = (1-q)^k \sum_{m=0}^\infty {m+k \choose k} q^m $$
 * See the binomial series article. — Fly by Night  ( talk )  21:45, 6 November 2011 (UTC)
 * Specifically the "special cases" section, from which you get $$\sum_{m=0}^\infty {m+k \choose k} q^m=(1-q)^{-k-1}$$ and so $$f_k(q)=\frac1{1-q}$$. -- Meni Rosenfeld (talk) 10:03, 7 November 2011 (UTC)


 * The inspiration for this question came from me trying to play with the binomial distribution $$ P(k,n,p) = {n\choose k}p^k(1-p)^{n-k}$$ but trying to make all the variables k,n and p random variables. So I noticed that I could integrate p from 0 to 1, and also sum from k = 0 to n and show that with k fixed, P still looks like a joint probability density. But once I try to sum from n = 0 infinity, this doesn't work. So I guess I want to know whether it is possible to make a joint probability density involving all the variables....is it actually possible to do something like this? --HappyCamper 04:37, 8 November 2011 (UTC)

Fermat's Last theorem
What was the proof of this theorem? can anyone explain this to an average fella? i saw the article that andrew wiles proved the theorem but no actual proof was in there, and i am also wondering how to prove a math theorem, i thought that you just use the formula, substitute the variables and if it works, poof its proven. — Preceding unsigned comment added by 203.112.82.1 (talk) 22:11, 6 November 2011 (UTC)
 * No we can't explain it to an average fella. I'd probably have to work a few years on it myself to understand it all completely. Why on earth would you be interested if you're asking what application perfect numbers have? And you'd definitely need to be interested to devote a chunk of your life to it. Have a look at Euclid's theorem for a good example of a proof. Dmcq (talk) 22:44, 6 November 2011 (UTC)
 * Thanks for the answer, but what is the connection with my question with perfect numbers and my curiosity with fermats theorem? i dont get it. — Preceding unsigned comment added by 203.112.82.1 (talk) 23:04, 6 November 2011 (UTC)
 * They're both maths. Perfect numbers is some fairly basic maths that people with a slight leaning to seeing the beauty of maths will appreciate. If you don't see it why on earth would you be interested in Fermat's last theorem? Dmcq (talk) 23:17, 6 November 2011 (UTC)
 * I can see the beauty of those numbers, i am just thinking that there must be reason why people spend their time working on something other that its just 'cool'. — Preceding unsigned comment added by 203.112.82.1 (talk) 23:31, 6 November 2011 (UTC)
 * You can prove that a formula is false by substituting in numbers and finding that it doesn't work, but to prove a formula is true, you would have to substitute in every possible number for which it claims to be true. Proofs of a formula usually involve algebra.  For a formula that claims to be true for positive integers, a method called Mathematical induction is often used.  People spend their time working on problems that interest them.  In the case of Fermat's Last Theorem, part of the challenge might have been "the world's most difficult problem" and the fact that some of the world's best minds had failed to find a full proof for 358 years (though many others contributed to the final proof).   D b f i r s   00:05, 7 November 2011 (UTC)
 * Enlighten me with the theorem, so just like the pythagorean theorem, it is already working, the problem is just there is no actual proof why it works? or atleast before andrew wiles. am i right? — Preceding unsigned comment added by 203.112.82.1 (talk) 00:28, 7 November 2011 (UTC)
 * Fermat's last theorem is not at all like Pythagoras' theorem. The latter is true for all values, but Fermat's claim is that no such numbers exist.    D b f i r s   16:37, 7 November 2011 (UTC)
 * Cool is quite good enough. Being paid is a bonus. Thankfully there is a good lot of uses for maths so there's no need to live in a garret and starve like a painter. Dmcq (talk) 00:11, 7 November 2011 (UTC)

Our article is Wiles' proof of Fermat's Last Theorem. -- 49.228.87.218 (talk) 03:09, 7 November 2011 (UTC)

Continous chaotic system with a closed-form solution
Does anyone know of a continous chaotic system with an explicit/closed-form solution? One that generates a strange attractor is preferable. I wish to experiment with initial conditions but still have exact results. --Melab±1 &#9742; 23:53, 6 November 2011 (UTC)
 * Any closed form solution would approach either a periodic a steady state, so by definition it would not be chaotic. Solutions can be estimated with numerical methods, but the error will tend to grow until the result is meaningless. You can delay this by using more accurate methods but the result will eventually be the same.--RDBury (talk) 00:15, 7 November 2011 (UTC)
 * Are you saying that any system with a closed form solution cannot be chaotic? If this extends to both discrete and continuous systems, a counter example to what you said can be found on Wolfram. --Melab±1 &#9742; 00:32, 7 November 2011 (UTC)
 * The map has a closed form but the solution does not. If you read the link you gave it says closed form solutions exist only for r=-2, r=2 and r=4.--RDBury (talk) 18:28, 7 November 2011 (UTC)
 * But they exist nonetheless and values where r equals -2, 2, or 4 exhibit chaotic behavior. --Melab±1 &#9742; 19:42, 7 November 2011 (UTC)
 * The three equations given on the Wolfram page are closed-form solutions that enable someone to find $$x_n$$ when given n. --Melab±1 &#9742; 23:43, 7 November 2011 (UTC)