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Old 2009-01-18, 13:19   #1
mart_r
 
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Default Mysterious connection

Hello again,

some time ago I noticed that there seems to be a strange connection between the standard normal distribution and 1/tanh(pi^2), as to be seen in A133658 in the OEIS: http://www.research.att.com/~njas/sequences/A133658

I wondered if any of you are able to tell me what's going on with these numbers.

Last fiddled with by mart_r on 2009-01-18 at 13:21
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Old 2009-01-25, 15:53   #2
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Is this even interesting? A coincidence? Or is it a trivial phenomenon?

Last fiddled with by mart_r on 2009-01-25 at 15:55
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Old 2009-01-25, 16:05   #3
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Quote:
Originally Posted by mart_r View Post
Is this even interesting? A coincidence? Or is it a trivial phenomenon?
It's like that one where e^(pi*sqrt(163)) is nearly an integer.
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Old 2009-10-22, 18:14   #4
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Had a look into this again today and found:

1/tanh(pi²) = 1 + 2*[e^(-2pi²)+e^(-4pi²)+e^(-6pi²)+e^(-8pi²)+...] (rather easy)
A133658 __= 1 + 2*[e^(-2pi²)+e^(-8pi²)+e^(-18pi²)+e^(-32pi²)+e^(-50pi²)+...] (rather interesting)

I would try to figure out why this is so, if only I could find an explanation of how the factor of 1/sqrt(2pi) in the standard normal distribution comes about...

(Come think of it, this thread should probably be moved to Math or MiscMath)

Last fiddled with by mart_r on 2009-10-22 at 18:16
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Old 2009-10-24, 18:17   #5
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Quote:
Originally Posted by mart_r View Post
I would try to figure out why this is so, if only I could find an explanation of how the factor of 1/sqrt(2pi) in the standard normal distribution comes about...
Are you familiar with the neat trick to find the area under the whole
curve e^(-ax^2)?
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Old 2009-10-24, 18:40   #6
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Quote:
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Are you familiar with the neat trick to find the area under the whole
curve e^(-ax^2)?
Nope. Not yet.
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Old 2009-10-24, 19:04   #7
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Hint:
Find the volume under the surface e^-a(x^2 + y^2)

Don't spoil the fun by googling
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Old 2009-10-24, 19:40   #8
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Assuming a sphere (which I guess is the wrong way, right?),

e^(-3a(x²+y²)/2)/6pi^(3/2)

(is there something like an instruction for use of the TEX-format?)

Last fiddled with by mart_r on 2009-10-24 at 19:41
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Old 2009-10-24, 20:16   #9
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Quote:
Originally Posted by davieddy View Post
Hint:
Find the volume under the surface e^-a(x^2 + y^2)

Don't spoil the fun by googling
I am also frustrated by my inability to script maths
as I do on paper.

z = e^-a(x^2 + y^2) is the surface
Find the the volume between it and the whole plane z=0.

PS Pythagoras comes in handy.

Last fiddled with by davieddy on 2009-10-24 at 20:30
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Old 2009-10-24, 20:20   #10
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http://www.mersenneforum.org/showthread.php?t=4576
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Old 2009-10-25, 11:10   #11
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Quote:
Originally Posted by davieddy View Post
z = e^-a(x^2 + y^2) is the surface
Find the the volume between it and the whole plane z=0.
Okay, I admit that I lack some experience in this area of mathematics.
(z is the surface? I thought it was the room coordinate?)

Maybe I'd figure it out eventually, but not today.

@ Mini-Geek: thanks!

Last fiddled with by mart_r on 2009-10-25 at 11:12
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