Re: Another integer-like transcendental number

Robin Chapman <[EMAIL PROTECTED]> wrote:
>TCL <[EMAIL PROTECTED]> wrote:
>> 
>> Let a = 1/10, b = sqrt(2)/20. Then
>>
>> cos(a)+cosh(a)+2cos(b)cosh(b) = 4.000000000000992...
>>
>> Can anyone explain this one? Or give a reference.
> 
> cos(a) + cosh(a) = 2 (1 + a^4/4! + a^8/8!   + ...)
> cos(b) * cosh(b) =    1 - b^4/6  + b^8/2520 + ... .
> 
> If  a^4 = 4 b^4  (as it is here) then
> cos(a) + cosh(a) + 2 cos(b)cosh(b) = 4 + b^8/1008 + ...
> is very close to 4 if b (and so a) is small.

Simpler: f(x) := 2 (cos(x) + cosh(x)) + cos(x) cosh(x) 

               = 5 + x^8/2016 + O(x^12)

e.g.    f(0.1) = 5.00000000000496 ~= 5 + 5*10^-12

For deeper numerical coincidences see my prior post below.
-------------
sci.math post of 1998/06/25: Subject: Re: Can e & pi be easily related? 
http://google.com/groups?threadm=y8zlnqlmzwl.fsf%40nestle.ai.mit.edu

Bill Taylor <[EMAIL PROTECTED]> wrote:
| 
| Is there a reasonably simple real-numbers equation connecting  
| e & pi, that can be used to calculate either one from the other?

One of the most beautiful and mysterious approximations is

        pi             3        1/sqrt(163)             -30
       e    =  ( 640320  + 744 )             -  5.177 10

Check it on your calculator, then see my prior post [1] 
for references to the deep and fascinating explanation
involving class fields, complex multiplication, modular 
functions, Kronecker's Jugendtraum (youthful dream), etc.

Or behold Shanks' [2] amazing approximation

               6              24  -r pi            -163
        pi  =  - log(2abcd) + -- e       +  6.69 10
               r               r

 a = A+sqrt(A^2-1), b = B+sqrt(B^2-1), c = C+sqrt(C^2-1), d = D+sqrt(D^2-1), 

 A = 1/2 (1071 + 184 sqrt(34)),  B = 1/2 (1553 + 266 sqrt(34))

 C = 429 + 304 sqrt(2),  D = 1/2 (627 + 442 sqrt(2)),  r = sqrt(3502)

To improve an approximation P to pi use  P + sin(P)  or  P + 2 cos(P/2),
which triples the accuracy, or use  P + (2 sin P - tan P)/3,  which 
quintuples the accuracy, cf. [3]. For example, applying these to the
 pi approximation from the first equation above triples the accuracy
from 10^-31 to 10^-93, and quintuples the accuracy to 10^-155.
Quintupling Shanks' approximation yields over 800 digits of pi.

-Bill Dubuque

[1] sci.math post of 1996/09/27 titled: Re: (pi+20)^i ~= -1, explain why
http://google.com/groups?selm=y8zohirmj49.fsf%40martigny.ai.mit.edu

[2] Shanks, Daniel. Dihedral quartic approximations and series for pi. 
J. Number Theory 14 (1982), no. 3, 397-423. 
MR 83k:12010 (Reviewer: Harvey Cohn) 12A70

[3] Shanks, Daniel.  Improving an approximation for pi. 
Amer. Math. Monthly 99 (1992), no. 3, 263. 
MR 94a:11197 (Reviewer: W. W. Adams) 11Y60
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