Physical proof of the Stirling formula

Русский вариант представленного ниже текста можно найти в журнале "Физическое образование в ВУЗах", 2018, №2 по имяреку автора. Журнал платный как для читателей, так и для авторов (!), а я публикацию не оплатил. Тем не менее, статью опубликовали, но - без присылки мне бумажных авторских экземпляров и даже без присылки *.pdf - файла с окончательным вариантом текста. Таким образом, на руках у меня только *.doc - файл исходной рукописи, который я охотно вышлю бесплатно любому написавшему мне по mail-адресу, указанному в самом конце текста. (Давать же здесь ещё и русский текст не хочется.)
 
[Pi = 3.14159, e = 2.718, П{Сk} = C1·C2·C3· ... ·Cn from number k = 1 to k = n, a**3 = a· a· a and so on,
"lim" is at n plus infinity elsewhere, "sqr" and "sqrt" are usual symbols of Turbo-pascal language,
"Д" is the greek "delta" symbol, the SI system is used,
"Log" is the natural logarithm function, "ch" is the hyperbolic cosine function]
 
1.    Introduction
 
In 1730, the scottish mathematician James Stirling publiced the book "METHODUS DIFFERENTIALES ... "
in which one can find the next result:
 
Log1 + Log2 + ... + Log(n)  = (n + 0.5)· Log (n + 0.5) + 0.5· Log(2 Pi) - (n + 0.5) - O(1/n)
 
Exponentiate it, we obtain the so-named Stirling formula
 
n! = П{k} = 1·2·3· ... ·n ~ sqrt(2·Pi)·((n + 0.5)/e)**(n + 0.5)    (1)
 
It seems, that there are no any course of mathematical analysis without (1) in the modern world and
numerous ways to prove (1) was presented in different lands by different authors, what has been
observed in [1]. But the Stirling's formula, equivalently submitted as
 
n! ~ sqrt(2·Pi·n)·(n/e)**n   (2)
 
is used not only in courses of mathematical analysis but also in university courses of theoretical
physics. For example, it appears when they study fluctuations of number of molecules in an open
volume of ideal gas ([2]). In some of physics textbooks there is no proof of (2). In [3],
it is presented along with the mathematical proof based on transformation of the Euler - Lagrange
formula for n! as definite integral of function exp(-t)·t**n from t = 0 to t - infinite,
which may be easy established by the mathematical induction method having however no any connection
with physics itself. Proof of the formula (2) in textbook on mathematics for physicists [4] has no
any connection with physics too. So the purpose of this note is to suggest a method of derivation
of (2) understandable to an undergraduate student of physics and closer to university course of
general physics compared to the published evidence of the formula (2) mentioned above.
The knowledge only of content of the first two chapters of textbook [5] is necessary for proof's
understanding besides any standard analysis course for physicists.
 
2.    The proof of formula (2)

Consider Oxy plane of the Cartesian coordinate system Oxyz divided into plates of equal width 2a
by straight lines Xk = 2k·a, where integer k numerates the plate to the right of the point Xk.
Plates are charged with electricity (set of charged isolated metal nano-particles lying as a thin
layer Z = 0 in paraffine matrix) and the k-th plate surface density of electrical charge is
 
S(r) = (So· X)/a - (2k + 1)·So        (3)

where r is the radius-vector of point of observer (not shown in fig.1),
So > 0 is a constant (figure 1) and Xk < X < X|k + 1|.
The Fourier expansion of this function is the sum of items 
 
- 2·So · sin((Pi · m · X)/a) / (Pi·m)         (4)
 
from m = 1 to m - infinity.
Vertical component Ez(r) of the electric field E(r) of such system of charges  corresponds to the next boundary condition ([5], e0 is the electrical permittivity of free space):
 
Ez(X, Z = +0) = S(X)/(2 · e0)              (5)
 
The system's electrostatical potential Ф(r) satisfies the Laplace equation
 
Фxx + Фzz = 0            (6)
 
The solution Ф(r) of the equation (6) with the boundary condition (5), where S(r) is defined by
(3) and Ф(r) connected with E(r) by identity E(r) = - grad Ф(r), can be obtained using the method
of separation of variables ([6], Pt. 10) and is the sum of items
 
 - [So· a · sin((Pi· m · x)/a ) · exp(-(Pi· m · z)/a) ] / [sqr(Pi· m)·e0 ]   (7)      
 
from m = 1 to m - infinity (z > 0).
Then the x-component value Ex of the electrical field E(r) сreated by the charge's system (4) in
the point А(X = 0, Z = Za) over the boundary between adjacent plates will be:
 
Ex(Za) = - dФ / dx = So · Log[1 - exp(- Pi· Za)/a )] / (Pi · e0)          (8)
 
where the Taylor’s formula for logarithmic function was used (some different methods of expansion
procedure for
 
Log[1 - 2·A·cosB + sqr(A)] at |A| < 1, |B| < 1
 
can be found in textbook [7]).
 
But the field vector E(r) can be calculated also as sum of fields created by individual plates on
figure 1 and k-th plate field ДEk in the point А, in turn, is derived by integrating fields
dE(x) of infinitely thin charged strips, parallel to Oy axis, on which we split this plate.
Module |dE(x)| of the stripe field vector dE(x) is easy calculated using the Gauss’s law ([5]),
integral is calculated elementarly and electrical field's x-projection is:
 
ДEk,x = (So / (Pi·e0))·       
 
·[(k + 0.5)·Log(R|k+1|/ Rk) - 1 + (0.5· Za / a)·(arctg(2(k + 1)· a / Za)  - arctg(2k· a / Za)]   (9)
 
where Rk = sqrt(sqr(Xk) + sqr(Za)).
Summing (9) over all "k", equating sum to right-side of (8), regrouping terms, cutting the common
factor So / (Pi·e0), introducing dimensionless variables g = Za / 2a , Hk = Rk / 2a,  we make play:
 
Ln[2(ch(2· Pi· g) - 1)] = 4·lim[n + Log П{(Hk / H|k+1|)**(0.5 + k)}]                (10)
 
(The sum of logarifms in the formula (10) which is equivalent logaritm of product, converges uniformly
([7]) by this, as its general term has asymptotic form:
 
(5 + 4 sqr(g)) / sqr(k) + O(k**(-3))      (11)               
 
). Simple algebraic transformations of the right side of the equation (10) result:
 
Log[2(ch(2· Pi· g) - 1)] - 2Log(g) = 4·lim[n + Log [ П{Hk} / (Hn)**(n + 0.5)]]                (12)
 
Supposing g = + 0 in (12), so that the point A in the figure 1 moves to the coordinate center О, we
obtain after calculation of limit in left-hand side of the identity (12):
 
Log[4· sqr(Pi)] = 4· Lim [n + (Log(n - 1)!) / n**(n - 0.5)]                (13)
 
It is, evidently, equivalent to (2). The proof is finished.
One can say now that the derivation of (2) may be considered as a typical electrostatic problem.
 
3. Addition
Further material lies out of the theoretical physics textbooks content but it may be interesting for
a student-matematician as example of unexpected connection between mathematics and physics.
What will be if we transform from point A to point B in figure 1?
Calculations become more complex and if we introduce f = xB / 2a, then we obtain
 
ch(2· Pi· g) - cos(2· Pi· f) = 2· sqr(Pi)· (sqr(f) + sqr(g))· П{sqr(Hk)·sqr(H(-k))/(k**4)}             (14)
 
where Hk = sqrt(sqr(Xk - Xb) + sqr(Za)) / 2a now.
All four L. Euler’s classical identities [6] for elementary functions expanding into infinite
products can be concluded from (14) namely:
the formula for expanding sin(Pi· f) / (Pi· f) at g = 0,
the formula for expanding sh(Pi· g) / (Pi· g) at f = 0,
the formula for expanding ch(2Pi· g) / (2Pi· g) at f = 0.25
and the Euler’s cosine product formula cos(Pi· f) / (Pi· f) is a simple consequence of the formula
for expanding sin(Pi· f) / (Pi· f) and the identity cosq = sin(Pi/2 – q)).
The analogous procedure of calculation of the field z–component Ez in space leads to formula what
generalized the well-known Bazel problem of XVII century: to calculate the sum of reversed squares of natural numbers [8].
This formula may be found in [9] (page 135, formula 6.707, example 2) but the way of proof there is other comparably with the presented here.
 
4. Literature

[1] J. M. Borwein, R. M. Corless, Amer. Math. Monthly. 121, 400 (2018).
[2] L. D. Landay, E. M. Lifshits, Statistical Physics – Pt 1. 3rd ed., (Pergamon Press, 2013).
[3] A. Isihara, Statistical Physics. 2rd ed., (Academic Press, 2013).
[4] Ya. B. Zeldovich, I. M. Yaglom, Higher math for beginners: mostly physicists and engineers, (Prentice-Hall, 1987)
[5] E. M. Purcell, D. J. Morin, Electricity and magnetism. 3rd ed., (Cambridge University Press, 2018).
[6] G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, (McGraw-Hill, 1968).
[7] G. M. Fikhtengol’ts The fundamentals of mathematical analysis, Vol II, (Pergamon, 1965).
[8] M. Ivan, General Mathematics 16, 111 (2008).
[9] E. P. Adams, Smitsonian mathematical formulae (Smitsonian Institution, 1922).
 
Author will be glad to receive any notes in connect with this text. My mail-adress:
 
shkhb4lsmp@mail.ru
 
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