A small numerical error of A. Einstein
- From: "David W.Cantrell" <DWCantrell@xxxxxxxxxxx>
- Date: Fri, 1 Feb 2008 18:31:29 +0000 (UTC)
I seem to have discovered a small numerical error in Einstein's
"Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung",
Sitzungsberichte der Preussischen Akademie der Wissenschaften,
Physikalisch-Mathematische Klasse, 1925, no. 1, pp. 3-14. The error
occurs near the bottom of page 13. In Einstein's manuscript, available
online, the error occurs at the top of page 15; see
<http://www.lorentz.leidenuniv.nl/history/Einstein_archive/Einstein_1925_manuscript/Pages/Einstein_1925_15a.html>.
The error is, I suppose, quite insignificant. Nonetheless, I'm curious
to know if it has been discovered previously.
For me (a mathematician), the tale started a few days ago in a newsgroup
devoted to the computer algebra system Mathematica. (If interested in
that thread, "Polylog equations", see
<http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_frm/thread/a98f725137c3eabb>.)
To explain things succinctly, with Li denoting polylogarithm, given y =
Li_{3/2}(lambda) and z = Li_{5/2}(lambda), we wish to obtain z as a
function of y. Presumably this cannot be done in closed form using known
functions, and so Einstein just gave the first few terms of the
Maclaurin series
z = y - 0.1768 y^2 - 0.0034 y^3 - 0.0005 y^4
as an approximation. But the latter two coefficients, if given to four
decimal places, should have been -0.0033 and -0.0001 instead.
Using Mathematica, it's very easy to get the first few terms of the
Maclaurin series precisely, and then to approximate the coefficients:
In[8]:= Normal[Simplify[
PolyLog[5/2, InverseSeries[Series[PolyLog[3/2, x], {x, 0, 6}], y]]]]
Out[8]= y - y^2/(4*Sqrt[2]) + (1/8 - 2/(9*Sqrt[3]))*y^3 +
((-18 - 15*Sqrt[2] + 16*Sqrt[6])*y^4)/192 +
(317/1728 + 1/(4*Sqrt[2]) - 1/(2*Sqrt[3]) - 4/(25*Sqrt[5]))*y^5 +
((-9450 - 8435*Sqrt[2] + 2400*Sqrt[3] + 4800*Sqrt[6] + 1728*Sqrt[10])*y^6)/34560
In[9]:= N[%]
Out[9]= y - 0.176777 y^2 - 0.00330006 y^3 - 0.000111289 y^4
- 3.5405*10^-6 y^5 - 8.38635*10^-8 y^6
Three other little comments on the last page of the published article:
1. The largest value of y considered should be zeta(3/2) = 2.612375...,
rather than Einstein's 2.615 .
2. In equation (18c), instead of N, we should have N^4. (The manuscript
is correct in this regard, but the exponent is hard to read.)
3. In the final equation, (22d), Einstein gives an approximate
coefficient, -0.186, representing the slope of a linear approximation. I
presume that Einstein intended to calculate that slope using the
endpoints of the graph of F(y). If that presumption is correct, then we
can easily obtain the desired slope precisely in terms of the Riemann
zeta function:
(zeta(5/2)/zeta(3/2) - 1)/zeta(3/2)
which is -0.186224... That agrees with Einstein's stated value of
-0.186, of course. But, considering his previous numerical errors, I
can't understand how he got -0.186 ! As best I can tell, he should have
gotten -0.189 instead.
David W. Cantrell
.
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