Let 
be a complex
number, then inequality
 |
(1)
|
holds in the lens-shaped region illustrated above. Written explicitly in terms of real variables, this can be written as
![1+lambda+sqrt(2(1+lambda-x^2+y^2))>exp[sqrt(2(1+lambda-x^2+y^2))],](/images/equations/LaplaceLimit/NumberedEquation2.svg) |
(2)
|
where
![lambda=sqrt([(1-x)^2+y^2][(1+x)^2+y^2]).](/images/equations/LaplaceLimit/NumberedEquation3.svg) |
(3)
|
The area enclosed is roughly
 |
(4)
|
(OEIS A140133).
This region can be parameterized in terms of a variable 
as
 |  |  |
(5)
|
 |  |  |
(6)
|
Written parametrically in terms of the Cartesian coordinates,
 |  |  |
(7)
|
 |  |  |
(8)
|
This region is intimately related to the study of Bessel functions and Kapteyn series (Plummer 1960, p. 47; Watson 1966, p. 270).
reaches its maximum value at 
(OEIS A085984;
Goursat 1959, p. 120; Le Lionnais 1983, p. 36), given by the root of
 |
(9)
|
or equivalently by the root of
 |
(10)
|
as noted by Stieltjes.
The minimum value of 
corresponding to the maximum value 
is 
(OEIS A033259;
Plummer 1960, p. 47; Watson 1966, p. 270), which is known as the Laplace
limit constant. It is precisely the point at which Laplace's formula for solving
Kepler's equation begins diverging, and is given
by the unique real solution 
of 
for
 |
(11)
|
The continued fraction of 
is given by [0, 1, 1, 1, 27, 1, 1, 1, 8, 2, 154, ...] (OEIS
A033260). The positions of the first occurrences
of 
in the continued
fraction of 
are 2, 10, 35, 13, 15, 32, 101, 9, ... (OEIS A033261).
The incrementally largest terms in the continued
fraction are 1, 27, 154, 1601, 2135, ... (OEIS A033262),
which occur at positions 2, 5, 11, 19, 1801, ... (OEIS A033263).
such that 
Using Library Functions?."
Mar. 12, 2022.