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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013



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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(-ax2) 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 sn=(n+1/2)Ln(n)-n-Ln(n!).
Let us study the convergence of this by letting un=sn-sn-1. Simplifying we get :

hence the serie un is convergent and hence so is the  sequence sn (since the term sn-1, sn-2... cancel each other by twos when we sum un).
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=t1/2, 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 Ca = {(x,y)R+2 | x2+y2a2} and the limits R2 Ka=[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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