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Topic: asymptotic expansion (Read 3089 times) |
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Moon
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Please help me with this problem Show that [int]0 [infty](1+t/k)kexp(-t)dt~1/2+1/(8k)-1/(32k2), k-> [infty]
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Icarus
wu::riddles Moderator Uberpuzzler
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Re: asymptotic expansion
« Reply #1 on: May 1st, 2004, 9:38am » |
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Is this a "method of steepest descents problem"?
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"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous
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Moon
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I think this is a Laplace's Method.
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Icarus
wu::riddles Moderator Uberpuzzler
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Re: asymptotic expansion
« Reply #3 on: May 1st, 2004, 8:40pm » |
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If you mean the Laplace transform, that may be the origin of this problem, but I don't think it offers any help in how to solve it. The problem as stated is not quite as bad as it seems. The limit of the right as k [to] [infty] is 1/2. Hence to show the desired asymptotic equivalence all you need to do is prove that the limit of the integral as k [to] [infty] is also 1/2. I.e., you don't need the 1/(8k) - 1/(32k2) to prove the asymptotic equivalence. This is an interesting integral. If you could exchange integral and limit, taking the limit of the function as k [to] [infty] yields [int]0[supinfty] ete-t dt, which is infinite. But such exchange of the integral and limit is not justified in this situation. Right now, the best I can suggest is to look at the method of steepest descents. It is used for producing such asymptotic formulas.
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"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous
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Icarus
wu::riddles Moderator Uberpuzzler
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Re: asymptotic expansion
« Reply #4 on: May 6th, 2004, 5:21pm » |
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You may or may not have noticed that I posted the integral here in the medium forum, where it would get more attention (I am not sure what other regulars visit this forum, but I seem to be the only one providing much in the way of answers in it.) T&B posted that version on sci.math, where for some reason it got better response than what I assume is your own post there. Several responders there and two on the other thread here demonstrated that the limit of the integral is [infty], not 1/2 as the other side of your asymptotic equivalence demands. The most comprehensive result was posted by David C. Ullrich on sci.math:limk k-1/2[int]0[supinfty](1 + t/k)ke-tdt = [surd]([pi]/2). [surd]([pi]k/2) is definitely not ~ 1/2 + 1/(8k) - 1/(32k2).
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"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous
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Alexey Sukhinin
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If you make a substitution 1+t/k = exp(u) and then integrate the result by parts you'll get the needed result.
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Icarus
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Re: asymptotic expansion
« Reply #6 on: May 7th, 2004, 6:56pm » |
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I may get the "needed" result but I certainly don't get the desired result! Let f(n) = [int]0[supinfty] (1+t/k)ne-t dt. I prefer the substitution x = k + t. The bookkeeping is slightly easier. The substitution gives f(n) = ekk-n[int]k[supinfty] xne-xdx.Integrate by parts (u=xn, v=-e-x) to get f(n)=ekk-n[ -xne-x]k[supinfty] +(n/k)ekk1-n [int]k[supinfty] xn-1e-x dx f(n) = 1 + (n/k)f(n-1). So, assuming k is an integer, f(k) = 1 + (k/k)(1 + ((k-1)/k)(1 + ((k-2)/k)(1 + ... + (1/k)f(0))...) Clearly f(0)=1, and expanding the expression above yields: f(k) = 1 + k/k + k(k-1)/k2 + k(k-1)(k-2)/k3 + ... + k!/kk = [sum]n=0k k!/(k-n)! k-n. This not only does not look like 1/2+1/(8k)-1/(32k2), but is easily seen not to be asymptotically equivalent: Each term in the sum = C(k,n)n!k-n > C(k,n)k-n. The sum of the latter terms is (1+1/k)k, which converges to e as k [to] [infty]. So whatever limit f(k) has as k [to] [infty], it has be [ge] e, but the limit of 1/2 + 1/(8k) - 1(32k2) is 1/2. In fact, as has been demonstrated in the calculations cited, limk[to][subinfty] f(k) = [infty]. I suppose that a different approach might actually give you f(k) = 1/2 + 1/(8k) - 1/(32k2) + g(k) for some function g. But the limit of g(k) as k [to] [infty] will not be zero, as the supposed asymptotic equivalence requires.
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« Last Edit: May 7th, 2004, 6:58pm by Icarus » |
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"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous
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