Factorial of an irrational number
Laurier St-Amant
amounts to?
Laurier St-Amant
irrational number?
Zdislav V. Kovarik
Laurier St-Amant <stam...@CollegeSherbrooke.qc.ca> wrote:
:Can someone tell me what the factorial of the square root of two (2)
:amounts to?
By the combinatorial definition: it's not defined (how do you re-arrange
2^(1/2) objects? How do you even obtain that many objects, no fewer and
no more?)
By an analytic (logarithmically convex) extension: Gamma(1+2^(1/2)), same as
2^(1/2)*Gamma(2^(1/2)), where for real part(s)>0,
Gamma(s) = integral(from 0 to inf) (x^(s-1)*e^(-x))dx
Cheers, ZVK (Slavek)
Claudius Mueller
says...
>
>Can someone tell me what the factorial of the squareroot of two (2)
>amounts to. Is it possible to define the factorial of a real or
>irrational number?
Hi,
Maybe I can help.
There are so many definition of factorial.
One of them is.
-t x-1
x! = gamma(x+1) gamma(x) = interg from 0 to inf e * t dt
Sure when x is real, it doesn't work
Try this:
x-1
n!*n
gamma(x) = lim ------------------------------------------
n runs to inf x(x-1)*(x+2).......(x+n-1)
and it works,
Propertyes of gamma:
gamma(1) = 1 gamma(x+1) = x*gamma(x)
gamma(0.5+ x)*gamma(0.5-x) = pi/( cos(pix))
And something for you
gamma(x)*gamma(-x) = -pi / ( x*sin(pi*x) )
gamma( sqrt(2) -1) * gamma ( -sqrt(2) + 1 ) =
= -pi / (- sqrt(2)* sin( sqrt(2)pi ) )
It is 'hardly' the same. (sorry)
I tried my best
Bye
>
jul...@ncgraphics.co.uk
>:amounts to?
My ancient Hewlett Packard 11C calculator evaluates funny factorials (of
non-integral numbers). For square root of two it gives 1.253815480
I've never actually used this feature for real, and think it's a bit silly
to put it in a calculator in those days when silicon was so expensive.
I wouldn't mind if someone found a new function other than the Gamma function
which evaluated to the same value as factorials for all integers, then people
would stop mistaking the two as being the same.
JT.
Eric Haas
>Can someone tell me what the factorial of the square root of two (2)
>amounts to?
Approximately 1.25381548064
Toby Bartels
>I wouldn't mind if someone found a new function other than the Gamma function
>which evaluated to the same value as factorials for all integers, then people
>would stop mistaking the two as being the same.
Gamma (x) cos (2 pi x)
-- Toby
to...@ugcs.caltech.edu
Martin Cohen
That's exactly the value I got on my HP-48!
Coincidence???
Dipankar Gupta
>
> >I wouldn't mind if someone found a new function other than the Gamma function
> >which evaluated to the same value as factorials for all integers, then people
> >would stop mistaking the two as being the same.
>
> Gamma (x) cos (2 pi x)
One with some historical significance can be written as:
1 d 1 - x
--------------- ---- log( Gamma( ----- ) / Gamma(1 - x/2) )
Gamma (1-x) dx 2
This function, due to Hadamard, has no singularities in the complex
plane, whereas Gamma does (at all negative integers).
--Dipankar
Dann Corbit
gamma(x) = (x-1)!
Calculators may either use the gamma function or Stirling's
approximation for the factorial, which does not need integer
input.
--
The opinions expressed in this message are my own personal views
and do not reflect the official views of Microsoft Corporation.
Rob Johnson
Laurier St-Amant <stam...@CollegeSherbrooke.qc.ca> wrote:
>Can someone tell me what the factorial of the squareroot of two (2)
>amounts to. Is it possible to define the factorial of a real or
>irrational number?
For an integer n, n! = Gamma(n+1) = n Gamma(n). Using the integral
representation of Gamma gives
|\oo -t n
n! = | e t dt [1]
\| 0
To see that [1] is true for an integer n, note that when n = 0, the
integral is 1. Next, integrate by parts to get
|\oo -t n |\oo n -t
| e t dt = - | t d(e )
\| 0 \| 0
n -t -+oo |\oo -t n
= -t e | + | e d(t )
-+0 \| 0
|\oo -t n-1
= 0 + | e nt dt
\| 0
|\oo -t n-1
= n | e t dt
\| 0
Thus, the integral in [1] holds for any non-negative integer n.
As for sqrt(2)!, there is Stirling's asymptotic series for Gamma:
In article <4b3n8p$i...@apple.com>,
I wrote:
>To the tenth order, Stirling's formula is
>
> x 1 1 139 571
> Gamma(x) = sqrt(2pi/x) (x/e) (1 + --- + ------ - -------- - ----------
> 12x 288x^2 51840x^3 2488320x^4
>
> 163879 5246819 534703531 4483131259
> + ------------ + -------------- - --------------- - -----------------
> 209018880x^5 75246796800x^6 902961561600x^7 86684309913600x^8
>
> 432261921612371 6232523202521089
> + --------------------- + ------------------------ - ...)
> 514904800886784000x^9 86504006548979712000x^10
>
>The error is less than the first odd term omitted.
Use Stirling's series to compute (98+sqrt(2))! and use the recursive relation
n! = n (n-1)! to reduce this to sqrt(2)! = 1.25381548064289168669637...
Of course, in the strictly combinatorial sense this makes no sense.
Rob Johnson
Apple Computer, Inc.
rjoh...@apple.com