wu :: forums « wu :: forums - power of cosine function » Welcome, Guest. Please Login or Register. Feb 8th, 2023, 6:32pm RIDDLES SITE WRITE MATH! Home Help Search Members Login Register
 wu :: forums    general    complex analysis (Moderators: SMQ, william wu, Icarus, Eigenray, ThudnBlunder, towr, Grimbal)    power of cosine function « Previous topic | Next topic »
 Pages: 1 Reply Notify of replies Send Topic Print
 Author Topic: power of cosine function  (Read 11619 times)
comehome1981
Newbie

Posts: 20
 power of cosine function   « on: Oct 25th, 2010, 8:40pm » Quote Modify

A question as follows:

It is clear that

[cos(k/n)]^(n^2)  ----> exp{-k^2/2} as n goes to infinity

Does this hold when k=n/2 or n or some fraction of n?

Another question:

Does one know the estimate of cos(x)  as x --> pi/2

Thanks for any tips!
 « Last Edit: Oct 26th, 2010, 12:21pm by comehome1981 » IP Logged
towr
wu::riddles Moderator
Uberpuzzler

Some people are average, some are just mean.

Gender:
Posts: 13730
 Re: power of cosine function   « Reply #1 on: Oct 26th, 2010, 1:17am » Quote Modify

on Oct 25th, 2010, 8:40pm, comehome1981 wrote:
 It is clear that     [cos(k/n)]^(n^2)  ----> exp{-k^2} as n goes to infinity
I've tried graphing it for a few k, and I must say, it's anything but clear. [cos(k/n)]^(n^2) / exp{-k^2} doesn't seem to convergence on 1.

Ah, wolframalpha says it should be exp(- 1/2 k^2)[/edit]

Quote:
 Does one know the estimate of cos(x)  as x --> pi/2
Isn't it just zero? cos(pi/2)=0
 « Last Edit: Oct 26th, 2010, 1:19am by towr » IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
pex
Uberpuzzler

Gender:
Posts: 880
 Re: power of cosine function   « Reply #2 on: Oct 26th, 2010, 1:32am » Quote Modify

on Oct 26th, 2010, 1:17am, towr wrote:
 Ah, wolframalpha says it should be exp(- 1/2 k^2)

Taking logs and using L'Hopital twice also gives this result
 IP Logged
pex
Uberpuzzler

Gender:
Posts: 880
 Re: power of cosine function   « Reply #3 on: Oct 26th, 2010, 1:36am » Quote Modify

on Oct 25th, 2010, 8:40pm, comehome1981 wrote:
 Does this hold when k=n/2 or n or some fraction of n?

No, it doesn't. If k = cn, clearly k/n = c and you're taking the limit of [cos(c)]n^2. This limit is obviously either 1 (if cos(c)=1), nonexistent (if cos(c)=-1), or zero (otherwise).
 IP Logged
towr
wu::riddles Moderator
Uberpuzzler

Some people are average, some are just mean.

Gender:
Posts: 13730
 Re: power of cosine function   « Reply #4 on: Oct 26th, 2010, 5:20am » Quote Modify

on Oct 26th, 2010, 1:36am, pex wrote:
 No, it doesn't. If k = cn, clearly k/n = c and you're taking the limit of [cos(c)]n^2. This limit is obviously either 1 (if cos(c)=1), nonexistent (if cos(c)=-1), or zero (otherwise).
But the limit of exp{- 1/2 k^2} = exp{- 1/2 (cn)^2} would also be 0. So the two expressions might still converge. (Were it not that as far as I can tell they generally don't.)

If you want
[cos(c)]n^2 -> exp{- 1/2 (cn)^2}
then you must have
cos(c) -> exp{- 1/2 c^2}
Since c is constant, the expressions on both sides have to be equal, and thus c must be 0. (Which means it falls under the original case, since k=cn is constant for c=0.)
[/edit]
 « Last Edit: Oct 26th, 2010, 5:29am by towr » IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
comehome1981
Newbie

Posts: 20
 Re: power of cosine function   « Reply #5 on: Oct 26th, 2010, 12:24pm » Quote Modify

For the second part, I meant

is there a formula for the error  bound for
|cos(x)|  when x closes to pi/2
 IP Logged
towr
wu::riddles Moderator
Uberpuzzler

Some people are average, some are just mean.

Gender:
Posts: 13730
 Re: power of cosine function   « Reply #6 on: Oct 26th, 2010, 12:28pm » Quote Modify

Sure, just use the one for -sin(x) at 0
 IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
pex
Uberpuzzler

Gender:
Posts: 880
 Re: power of cosine function   « Reply #7 on: Oct 27th, 2010, 12:31am » Quote Modify

on Oct 26th, 2010, 5:20am, towr wrote:
 If you want   [cos(c)]n^2 -> exp{- 1/2 (cn)^2} then you must have cos(c) -> exp{- 1/2 c^2} Since c is constant, the expressions on both sides have to be equal, and thus c must be 0. (Which means it falls under the original case, since k=cn is constant for c=0.)

I don't think we can simply cancel the exponent n2 when throwing infinities around: if you just want both sides to tend to zero, it is sufficient that c is not a multiple of pi. (But that's not something I would use the "->" notation for...)
 IP Logged
towr
wu::riddles Moderator
Uberpuzzler

Some people are average, some are just mean.

Gender:
Posts: 13730
 Re: power of cosine function   « Reply #8 on: Oct 27th, 2010, 12:49am » Quote Modify

Well, they do both tend to zero, generally; but I figure we're interested in asymptotic behaviour, i.e. that one function approaches the other as n gets larger (rather than that they both approach a common limit).
So in that case, their quotient should tend to 1; and then it seems to me you can just cancel the n2 factors, because they don't qualitatively change the asymptotic behaviour.
If a(x)x/b(x)x goes to 1 as x increases, then a(x)/b(x) must go to 1 as x increases (a necessary, but not sufficient condition). But in our case a(x) and b(x) are constants, and if they're not equal, then the expression goes to 0 or +/-infinity.
 « Last Edit: Oct 27th, 2010, 12:56am by towr » IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
comehome1981
Newbie

Posts: 20
 Re: power of cosine function   « Reply #9 on: Oct 27th, 2010, 9:18pm » Quote Modify

Thanks for all you who replied.