Abraham de Moivre (1667-1754) / James
Stirling (1692-1770)
Fundamental...
Slices of his life
Abraham de Moivre was born in 1667 in
Vitry-le-François, but came to live in London at 18. The Nante
edict was just revoked and Abraham is son of Huguenots. He so earn his
life by using his quick and brilliant mind in pubs. He start studying
mathematics after reading the Principia by Newton
which he loved ! Friends with Newton and
member of the principle european academie (Royal Society, Paris,
Berlin...), he unfortunatly needs to give private lessons because,
being French, he can't teach in Engalnd in universities...
De Moivre iis without doubt one of the first to be interested in
applied maths in numberous domains. Probability, of course, with his
studies in the density of the normal distribution (exp(-ax^{2})
and the famous formula
associated with it), but also finances and demography !
A complete mathematician whose discovery in 1730 of what is commonly
called Stirling's formula (yeah, another abusse of names!) is only the
surface of his succes!
De Moivre was also interested in complex numbers with the formula that
bears his name.
But I also choose to put Stirling on this page because it is him in the
end who made popular the formula that bear his name. He discovered it
under the form ln(n!) the same year as De Moivre and he generalised it
by pushing the expansion as far as we can (by the serie of the second
formula) There remains no portrait of the poor Stirling who was born in
1692 in Garden (Scotland) and did his studies at Oxford. He taught in
Venice from 1715 to 1725, then in London starting from 1726, and was
mostly interested in curves and asymptotics calculus.
The drawing at the top of the page is his coat of arm.
Those two mathematicians prove how false is the often acclaimed idea
(now and back then) that mathematicians do not leave their den and know
nothing of their european neighbour's work!
Around
Abraham de Moivre and Stirling have hence,
as mentioned above, both found the famous formula at the top in 1730,
De Moivre did add to it the density calculation of a normal
distribution.
Those two result are fundamental in loads of domains and I use them a
bit everywhere in those pages. Anyway it is easy to notice that to a
certain extend, there is some kind of equivalence betwenn Stirling's
formula and Wallis' formula and that the link between the two formula
above is simply called the Gamma function!
I love so much this result ,
which, in the end is nearly as beautiful as
exp(i)=-1, that I offer here no less than two
proves!
Proves
Let us start with Striling's formulua so to
get rid of this classical analysis result...
For this let us introduce the sequence s_{n}=(n+1/2)Ln(n)-n-Ln(n!).
Let us study the convergence of this by letting u_{n}=s_{n}-s_{n-1}.
Simplifying we get :
hence the serie u_{n} is convergent and hence so is the
sequence s_{n} (since the term s_{n-1}, s_{n-2}...
cancel each other by twos when we sum u_{n}).
So :
But according to Wallis formula, we have : hence
Cool ! that's what we wanted !
Never the less, I haven't fond the prove for the general formula. I
only know that it comes from the logarithm application of a function
expansion (see Encyclopédie Universalis, if my memory serves me
right)
Now let us look at the density of a normal distribution or, since it's
the same, the area under the 'Gauss Bell'. Note that with the variable
change , x=t^{1/2}, on R^{+},
we will notice that it comes to the same as determining (1/2) where is Euler's famous
gamma function defined to be: (x)=.
1) First prove: I leave Camille Jordan
to speak ( analysis lesson at Polytechnique, 1st year 1892-1893)
:
2) Second method: the big
classical of a variable change !
Consider the set C_{a} = {(x,y)R^{+2} | x^{2}+y^{2}a^{2}} and the limits R^{2} K_{a}=[0,a]^{2}.
Now, let us introduce the well defined integral (do I really need to
prove it?) :
(sorry, the integral is not very esthetically pleasing, but the
copy-paste from the eqaution editor obviously does not work very well!)
The exponential function is positive and since
Then, we consider the polar coordinate transformation whose well known Jacobian is r.
By appling this change of variable to the double integral, we write :
Hence the bounds give us :
which give the expected result.
Note that the two different notation of the result, as just before
(Integral on R^{+}), and at the top of the page(integral on R)
are equivalent since the exponential term is even
Trials
The Moivre/Stirling formula is not a
model for performance... but it is so useful in probability and
analysis that we won't complain about it! And the generalisation of the
formula improves a bit the performance by pushing k further than 0 in
the sum.
This is waht we get for k=5 :
Let us see the trials :
n/k |
k=0 |
k=1 |
k=2 |
k=3 |
k=4 |
k=5 |
n=5 |
3,247 (0) |
3,1414 (3) |
3,141594 (5) |
7 decimals. |
8 decimals. |
9 decimals. |
n=10 |
3,1943 (1) |
3,14157 (4) |
3,1415927 (6) |
9 decimals. |
10 decimals. |
12 decimals. |
n=50 |
3,152 (1) |
7 decimals. |
9 decimals. |
13 decimals. |
16 decimals. |
20 decimals. |
n=100 |
3,1468 (2) |
7 decimals. |
10 decimals. |
16 decimals. |
20 decimals. |
23 decimals. |
n=200 |
3,144 (2) |
8 decimals. |
12 decimals. |
18 decimals. |
22 decimals. |
26 decimals. |
As much as extra terms bring a certain efficient at the beginning,
rapidly the sequence slows down and nothing stop the logarithm
convergence.
We can estimate this convergence :
k=0 |
k=1 |
k=2 |
k=3 |
k=4 |
k=5 |
Log(n) |
3.9Log(n) |
5.2Log(n) |
8Log(n) |
9,6Log(n) |
11.5Log(n) |
Myeah, not great... And with no help, the Delta2 by
Aitken is completly useless, oh well!
Note that we know that sum and integrals are not that distance in
mathematics (it's in fact the same object in Lebesgue Integral theory,
since Reimann's Integral correspond nearly to Lebesgue's measure, and
the sum to the one obtain by measuring the counting!)
It is then not difficult to understand that when we consider :
p is a number near . Well, very near even since p and are equal on the first 42 billion decimals according to a
prove by the Borwein. Which show us that we
can't come to a conclusion too quickly after calculating a few decimals
of a formula! Never the less, we can think that this kind of formula,
if there was no exponential, could be interesting to explout since we
just need to do an approching calculation of the integral to find the
decimal of Pi
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