Number, approximately 0.916
In
mathematics , Catalan's constant G , is defined by
G
=
β
(
2
)
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)
2
=
1
1
2
−
1
3
2
+
1
5
2
−
1
7
2
+
1
9
2
−
⋯
,
{\displaystyle G=\beta (2)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-\cdots ,}
where β is the
Dirichlet beta function . Its numerical value
[1] is approximately (sequence
A006752 in the
OEIS )
G = 0.915965 594 177 219 015 054 603 514 932 384 110 774 …
Unsolved problem in mathematics :
Is Catalan's constant irrational? If so, is it transcendental?
It is not known whether G is
irrational , let alone
transcendental .
[2] G has been called "arguably the most basic constant whose irrationality and transcendence (though strongly
suspected) remain unproven".
[3]
Catalan's constant was named after
Eugène Charles Catalan , who found quickly-converging series for its calculation and published a memoir on it in 1865.
[4]
[5]
Uses
In
low-dimensional topology , Catalan's constant is 1/4 of the volume of an
ideal hyperbolic
octahedron , and therefore 1/4 of the
hyperbolic volume of the complement of the
Whitehead link .
[6] It is 1/8 of the volume of the complement of the
Borromean rings .
[7]
In
combinatorics and
statistical mechanics , it arises in connection with counting
domino tilings ,
[8]
spanning trees ,
[9] and
Hamiltonian cycles of
grid graphs .
[10]
In
number theory , Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form
n
2
+
1
{\displaystyle n^{2}+1}
according to
Hardy and Littlewood's Conjecture F . However, it is an unsolved problem (one of
Landau's problems ) whether there are even infinitely many primes of this form.
[11]
Catalan's constant also appears in the calculation of the
mass distribution of
spiral galaxies .
[12]
[13]
Known digits
The number of known digits of Catalan's constant G has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.
[14]
Number of known decimal digits of Catalan's constant G
Date
Decimal digits
Computation performed by
1832
16
Thomas Clausen
1858
19
Carl Johan Danielsson Hill
1864
14
Eugène Charles Catalan
1877
20
James W. L. Glaisher
1913
32
James W. L. Glaisher
1990
20000
Greg J. Fee
1996
50000
Greg J. Fee
August 14, 1996
100000
Greg J. Fee &
Simon Plouffe
September 29, 1996
300000
Thomas Papanikolaou
1996
1500 000
Thomas Papanikolaou
1997
3379 957
Patrick Demichel
January 4, 1998
12500 000
Xavier Gourdon
2001
100000 500
Xavier Gourdon & Pascal Sebah
2002
201000 000
Xavier Gourdon & Pascal Sebah
October 2006
5000 000 000
Shigeru Kondo & Steve Pagliarulo
[15]
August 2008
10000 000 000
Shigeru Kondo & Steve Pagliarulo
[14]
January 31, 2009
15510 000 000
Alexander J. Yee & Raymond Chan
[16]
April 16, 2009
31026 000 000
Alexander J. Yee & Raymond Chan
[16]
June 7, 2015
200000 001 100
Robert J. Setti
[17]
April 12, 2016
250000 000 000
Ron Watkins
[17]
February 16, 2019
300000 000 000
Tizian Hanselmann
[17]
March 29, 2019
500000 000 000
Mike A & Ian Cutress
[17]
July 16, 2019
600000 000 100
Seungmin Kim
[18]
[19]
September 6, 2020
1000 000 001 337
Andrew Sun
[20]
March 9, 2022
1200 000 000 100
Seungmin Kim
[20]
Integral identities
As Seán Stewart writes, "There is a rich and seemingly endless source of definite integrals that
can be equated to or expressed in terms of Catalan's constant."
[21] Some of these expressions include:
G
=
−
1
π
i
∫
0
π
2
ln
ln
tan
x
ln
tan
x
d
x
G
=
∬
0
,
1
2
1
1
+
x
2
y
2
d
x
d
y
G
=
∫
0
1
∫
0
1
−
x
1
1
−
x
2
−
y
2
d
y
d
x
G
=
∫
1
∞
ln
t
1
+
t
2
d
t
G
=
−
∫
0
1
ln
t
1
+
t
2
d
t
G
=
1
2
∫
0
π
2
t
sin
t
d
t
G
=
∫
0
π
4
ln
cot
t
d
t
G
=
1
2
∫
0
π
2
ln
(
sec
t
+
tan
t
)
d
t
G
=
∫
0
1
arccos
t
1
+
t
2
d
t
G
=
∫
0
1
arcsinh
t
1
−
t
2
d
t
G
=
1
2
∫
0
∞
arctan
t
t
1
+
t
2
d
t
G
=
1
2
∫
0
1
arctanh
t
1
−
t
2
d
t
G
=
∫
0
∞
arccot
e
t
d
t
G
=
1
4
∫
0
π
2
/
4
csc
t
d
t
G
=
1
16
(
π
2
+
4
∫
1
∞
arccsc
2
t
d
t
)
G
=
1
2
∫
0
∞
t
cosh
t
d
t
G
=
π
2
∫
1
∞
(
t
4
−
6
t
2
+
1
)
ln
ln
t
(
1
+
t
2
)
3
d
t
G
=
1
2
∫
0
∞
arcsin
(
sin
t
)
t
d
t
G
=
1
+
lim
α
→
1
−
{
∫
0
α
(
1
+
6
t
2
+
t
4
)
arctan
t
t
(
1
−
t
2
)
2
d
t
+
2
artanh
α
−
π
α
1
−
α
2
}
G
=
1
−
1
8
∬
R
2
x
sin
(
2
x
y
/
π
)
(
x
2
+
π
2
)
cosh
x
sinh
y
d
x
d
y
G
=
∫
0
∞
∫
0
∞
x
4
(
x
y
−
1
)
(
x
+
1
)
2
y
4
(
y
+
1
)
2
log
(
x
y
)
d
x
d
y
{\displaystyle {\begin{aligned}G&=-{\frac {1}{\pi i}}\int _{0}^{\frac {\pi }{2}}\ln \ln \tan x\ln \tan x\,dx\\[3pt]G&=\iint _{[0,1]^{2}}\!{\frac {1}{1+x^{2}y^{2}}}\,dx\,dy\\[3pt]G&=\int _{0}^{1}\int _{0}^{1-x}{\frac {1}{1-x^{2}-y^{2}}}\,dy\,dx\\[3pt]G&=\int _{1}^{\infty }{\frac {\ln t}{1+t^{2}}}\,dt\\[3pt]G&=-\int _{0}^{1}{\frac {\ln t}{1+t^{2}}}\,dt\\[3pt]G&={\frac {1}{2}}\int _{0}^{\frac {\pi }{2}}{\frac {t}{\sin t}}\,dt\\[3pt]G&=\int _{0}^{\frac {\pi }{4}}\ln \cot t\,dt\\[3pt]G&={\frac {1}{2}}\int _{0}^{\frac {\pi }{2}}\ln \left(\sec t+\tan t\right)\,dt\\[3pt]G&=\int _{0}^{1}{\frac {\arccos t}{\sqrt {1+t^{2}}}}\,dt\\[3pt]G&=\int _{0}^{1}{\frac {\operatorname {arcsinh} t}{\sqrt {1-t^{2}}}}\,dt\\[3pt]G&={\frac {1}{2}}\int _{0}^{\infty }{\frac {\operatorname {arctan} t}{t{\sqrt {1+t^{2}}}}}\,dt\\[3pt]G&={\frac {1}{2}}\int _{0}^{1}{\frac {\operatorname {arctanh} t}{\sqrt {1-t^{2}}}}\,dt\\[3pt]G&=\int _{0}^{\infty }\operatorname {arccot} e^{t}\,dt\\[3pt]G&={\frac {1}{4}}\int _{0}^{{\pi ^{2}}/{4}}\csc {\sqrt {t}}\,dt\\[3pt]G&={\frac {1}{16}}\left(\pi ^{2}+4\int _{1}^{\infty }\operatorname {arccsc} ^{2}t\,dt\right)\\[3pt]G&={\frac {1}{2}}\int _{0}^{\infty }{\frac {t}{\cosh t}}\,dt\\[3pt]G&={\frac {\pi }{2}}\int _{1}^{\infty }{\frac {\left(t^{4}-6t^{2}+1\right)\ln \ln t}{\left(1+t^{2}\right)^{3}}}\,dt\\[3pt]G&={\frac {1}{2}}\int _{0}^{\infty }{\frac {\arcsin \left(\sin t\right)}{t}}\,dt\\[3pt]G&=1+\lim _{\alpha \to {1^{-}}}\!\left\{\int _{0}^{\alpha }\!{\frac {\left(1+6t^{2}+t^{4}\right)\arctan {t}}{t\left(1-t^{2}\right)^{2}}}\,dt+2\operatorname {artanh} {\alpha }-{\frac {\pi \alpha }{1-\alpha ^{2}}}\right\}\\[3pt]G&=1-{\frac {1}{8}}\iint _{\mathbb {R} ^{2}}\!\!{\frac {x\sin \left(2xy/\pi \right)}{\,\left(x^{2}+\pi ^{2}\right)\cosh x\sinh y\,}}\,dx\,dy\\[3pt]G&=\int _{0}^{\infty }\int _{0}^{\infty }{\frac {{\sqrt[{4}]{x}}\left({\sqrt {x}}{\sqrt {y}}-1\right)}{(x+1)^{2}{\sqrt[{4}]{y}}(y+1)^{2}\log(xy)}}dxdy\end{aligned}}}
where the last three formulas are related to
Malmsten's integrals.
[22]
If K(k ) is the
complete elliptic integral of the first kind , as a function of the elliptic modulus k , then
G
=
1
2
∫
0
1
K
(
k
)
d
k
{\displaystyle G={\tfrac {1}{2}}\int _{0}^{1}\mathrm {K} (k)\,dk}
If E(k ) is the
complete elliptic integral of the second kind , as a function of the elliptic modulus k , then
G
=
−
1
2
+
∫
0
1
E
(
k
)
d
k
{\displaystyle G=-{\tfrac {1}{2}}+\int _{0}^{1}\mathrm {E} (k)\,dk}
With the
gamma function Γ(x + 1) = x !
G
=
π
4
∫
0
1
Γ
(
1
+
x
2
)
Γ
(
1
−
x
2
)
d
x
=
π
2
∫
0
1
2
Γ
(
1
+
y
)
Γ
(
1
−
y
)
d
y
{\displaystyle {\begin{aligned}G&={\frac {\pi }{4}}\int _{0}^{1}\Gamma \left(1+{\frac {x}{2}}\right)\Gamma \left(1-{\frac {x}{2}}\right)\,dx\\&={\frac {\pi }{2}}\int _{0}^{\frac {1}{2}}\Gamma (1+y)\Gamma (1-y)\,dy\end{aligned}}}
The integral
G
=
Ti
2
(
1
)
=
∫
0
1
arctan
t
t
d
t
{\displaystyle G=\operatorname {Ti} _{2}(1)=\int _{0}^{1}{\frac {\arctan t}{t}}\,dt}
is a known special function, called the
inverse tangent integral , and was extensively studied by
Srinivasa Ramanujan .
Relation to other special functions
G appears in values of the second
polygamma function , also called the
trigamma function , at fractional arguments:
ψ
1
(
1
4
)
=
π
2
+
8
G
ψ
1
(
3
4
)
=
π
2
−
8
G
.
{\displaystyle {\begin{aligned}\psi _{1}\left({\tfrac {1}{4}}\right)&=\pi ^{2}+8G\\\psi _{1}\left({\tfrac {3}{4}}\right)&=\pi ^{2}-8G.\end{aligned}}}
Simon Plouffe gives an infinite collection of identities between the trigamma function, π 2 and Catalan's constant; these are expressible as paths on a graph.
Catalan's constant occurs frequently in relation to the
Clausen function , the
inverse tangent integral , the
inverse sine integral , the
Barnes G -function , as well as integrals and series summable in terms of the aforementioned functions.
As a particular example, by first expressing the
inverse tangent integral in its closed form – in terms of Clausen functions – and then expressing those Clausen functions in terms of the Barnes G -function, the following expression is obtained (see
Clausen function for more):
G
=
4
π
log
(
G
(
3
8
)
G
(
7
8
)
G
(
1
8
)
G
(
5
8
)
)
+
4
π
log
(
Γ
(
3
8
)
Γ
(
1
8
)
)
+
π
2
log
(
1
+
2
2
(
2
−
2
)
)
.
{\displaystyle G=4\pi \log \left({\frac {G\left({\frac {3}{8}}\right)G\left({\frac {7}{8}}\right)}{G\left({\frac {1}{8}}\right)G\left({\frac {5}{8}}\right)}}\right)+4\pi \log \left({\frac {\Gamma \left({\frac {3}{8}}\right)}{\Gamma \left({\frac {1}{8}}\right)}}\right)+{\frac {\pi }{2}}\log \left({\frac {1+{\sqrt {2}}}{2\left(2-{\sqrt {2}}\right)}}\right).}
If one defines the Lerch transcendent Φ(z ,s ,α ) (related to the
Lerch zeta function ) by
Φ
(
z
,
s
,
α
)
=
∑
n
=
0
∞
z
n
(
n
+
α
)
s
,
{\displaystyle \Phi (z,s,\alpha )=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+\alpha )^{s}}},}
then
G
=
1
4
Φ
(
−
1
,
2
,
1
2
)
.
{\displaystyle G={\tfrac {1}{4}}\Phi \left(-1,2,{\tfrac {1}{2}}\right).}
Quickly converging series
The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:
G
=
3
∑
n
=
0
∞
1
2
4
n
(
−
1
2
(
8
n
+
2
)
2
+
1
2
2
(
8
n
+
3
)
2
−
1
2
3
(
8
n
+
5
)
2
+
1
2
3
(
8
n
+
6
)
2
−
1
2
4
(
8
n
+
7
)
2
+
1
2
(
8
n
+
1
)
2
)
−
−
2
∑
n
=
0
∞
1
2
12
n
(
1
2
4
(
8
n
+
2
)
2
+
1
2
6
(
8
n
+
3
)
2
−
1
2
9
(
8
n
+
5
)
2
−
1
2
10
(
8
n
+
6
)
2
−
1
2
12
(
8
n
+
7
)
2
+
1
2
3
(
8
n
+
1
)
2
)
{\displaystyle {\begin{aligned}G&=3\sum _{n=0}^{\infty }{\frac {1}{2^{4n}}}\left(-{\frac {1}{2(8n+2)^{2}}}+{\frac {1}{2^{2}(8n+3)^{2}}}-{\frac {1}{2^{3}(8n+5)^{2}}}+{\frac {1}{2^{3}(8n+6)^{2}}}-{\frac {1}{2^{4}(8n+7)^{2}}}+{\frac {1}{2(8n+1)^{2}}}\right)-\\&\qquad -2\sum _{n=0}^{\infty }{\frac {1}{2^{12n}}}\left({\frac {1}{2^{4}(8n+2)^{2}}}+{\frac {1}{2^{6}(8n+3)^{2}}}-{\frac {1}{2^{9}(8n+5)^{2}}}-{\frac {1}{2^{10}(8n+6)^{2}}}-{\frac {1}{2^{12}(8n+7)^{2}}}+{\frac {1}{2^{3}(8n+1)^{2}}}\right)\end{aligned}}}
and
G
=
π
8
log
(
2
+
3
)
+
3
8
∑
n
=
0
∞
1
(
2
n
+
1
)
2
(
2
n
n
)
.
{\displaystyle G={\frac {\pi }{8}}\log \left(2+{\sqrt {3}}\right)+{\frac {3}{8}}\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}{\binom {2n}{n}}}}.}
The theoretical foundations for such series are given by Broadhurst, for the first formula,
[23] and Ramanujan, for the second formula.
[24] The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.
[25]
[26] Using these series, calculating Catalan's constant is now about as fast as calculating
Apery's constant ,
ζ
(
3
)
{\displaystyle \zeta (3)}
.
[27]
Other quickly converging series, due to Guillera and Pilehrood and employed by the
y-cruncher software, include:
[27]
G
=
1
2
∑
k
=
0
∞
(
−
8
)
k
(
3
k
+
2
)
(
2
k
+
1
)
3
(
2
k
k
)
3
{\displaystyle G={\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {(-8)^{k}(3k+2)}{(2k+1)^{3}{\binom {2k}{k}}^{3}}}}
G
=
1
64
∑
k
=
1
∞
256
k
(
580
k
2
−
184
k
+
15
)
k
3
(
2
k
−
1
)
(
6
k
3
k
)
(
6
k
4
k
)
(
4
k
2
k
)
{\displaystyle G={\frac {1}{64}}\sum _{k=1}^{\infty }{\frac {256^{k}(580k^{2}-184k+15)}{k^{3}(2k-1){\binom {6k}{3k}}{\binom {6k}{4k}}{\binom {4k}{2k}}}}}
G
=
−
1
1024
∑
k
=
1
∞
(
−
4096
)
k
(
45136
k
4
−
57184
k
3
+
21240
k
2
−
3160
k
+
165
)
k
3
(
2
k
−
1
)
3
(
(
2
k
)
!
6
(
3
k
)
!
3
k
!
3
(
6
k
)
!
3
)
{\displaystyle G=-{\frac {1}{1024}}\sum _{k=1}^{\infty }{\frac {(-4096)^{k}(45136k^{4}-57184k^{3}+21240k^{2}-3160k+165)}{k^{3}(2k-1)^{3}}}\left({\frac {(2k)!^{6}(3k)!^{3}}{k!^{3}(6k)!^{3}}}\right)}
All of these series have
time complexity
O
(
n
log
(
n
)
3
)
{\displaystyle O(n\log(n)^{3})}
.
[27]
Continued fraction
G can be expressed in the following form
[28]
G
=
1
1
+
1
4
8
+
3
4
16
+
5
4
24
+
7
4
32
+
9
4
40
+
⋱
{\displaystyle G={\cfrac {1}{1+{\cfrac {1^{4}}{8+{\cfrac {3^{4}}{16+{\cfrac {5^{4}}{24+{\cfrac {7^{4}}{32+{\cfrac {9^{4}}{40+\ddots }}}}}}}}}}}}}
The simple continued fraction is given by
[29]
G
=
1
1
+
1
10
+
1
1
+
1
8
+
1
1
+
1
88
+
⋱
{\displaystyle G={\cfrac {1}{1+{\cfrac {1}{10+{\cfrac {1}{1+{\cfrac {1}{8+{\cfrac {1}{1+{\cfrac {1}{88+\ddots }}}}}}}}}}}}}
This continued fraction would have infinite terms if and only if
G
{\displaystyle G}
is irrational, which is still unresolved.
See also
References
^ Papanikolaou, Thomas (March 1997).
Catalan's Constant to 1,500,000 Places – via Gutenberg.org.
^ Nesterenko, Yu. V. (January 2016), "On Catalan's constant", Proceedings of the Steklov Institute of Mathematics , 292 (1): 153–170,
doi :
10.1134/s0081543816010107 ,
S2CID
124903059 .
^ Bailey, David H.; Borwein, Jonathan M.; Mattingly, Andrew; Wightwick, Glenn (2013), "The computation of previously inaccessible digits of
π
2
{\displaystyle \pi ^{2}}
and Catalan's constant",
Notices of the American Mathematical Society , 60 (7): 844–854,
doi :
10.1090/noti1015 ,
MR
3086394
^
Goldstein, Catherine (2015),
"The mathematical achievements of Eugène Catalan" , Bulletin de la Société Royale des Sciences de Liège , 84 : 74–92,
MR
3498215
^
Catalan, E. (1865), "Mémoire sur la transformation des séries et sur quelques intégrales définies", Ers, Publiés Par l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. Collection in 4 , Mémoires de l'Académie royale des sciences, des lettres et des beaux-arts de Belgique (in French), 33 , Brussels,
hdl :
2268/193841
^
Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds",
Proceedings of the American Mathematical Society , 138 (10): 3723–3732,
arXiv :
0804.0043 ,
doi :
10.1090/S0002-9939-10-10364-5 ,
MR
2661571 ,
S2CID
2016662 .
^
William Thurston (March 2002),
"7. Computation of volume" (PDF) ,
The Geometry and Topology of Three-Manifolds , p. 165,
archived (PDF) from the original on 2011-01-25
^
Temperley, H. N. V. ;
Fisher, Michael E. (August 1961), "Dimer problem in statistical mechanics—an exact result",
Philosophical Magazine , 6 (68): 1061–1063,
Bibcode :
1961PMag....6.1061T ,
doi :
10.1080/14786436108243366
^ Wu, F. Y. (1977), "Number of spanning trees on a lattice", Journal of Physics , 10 (6): L113–L115,
Bibcode :
1977JPhA...10L.113W ,
doi :
10.1088/0305-4470/10/6/004 ,
MR
0489559
^
Kasteleyn, P. W. (1963), "A soluble self-avoiding walk problem",
Physica , 29 (12): 1329–1337,
Bibcode :
1963Phy....29.1329K ,
doi :
10.1016/S0031-8914(63)80241-4 ,
MR
0159642
^
Shanks, Daniel (1959), "A sieve method for factoring numbers of the form
n
2
+
1
{\displaystyle n^{2}+1}
", Mathematical Tables and Other Aids to Computation , 13 : 78–86,
doi :
10.2307/2001956 ,
JSTOR
2001956 ,
MR
0105784
^ Wyse, A. B.; Mayall, N. U. (January 1942), "Distribution of Mass in the Spiral Nebulae Messier 31 and Messier 33.",
The Astrophysical Journal , 95 : 24–47,
Bibcode :
1942ApJ....95...24W ,
doi :
10.1086/144370
^ van der Kruit, P. C. (March 1988), "The three-dimensional distribution of light and mass in disks of spiral galaxies.",
Astronomy & Astrophysics , 192 : 117–127,
Bibcode :
1988A&A...192..117V
^
a
b Gourdon, X.; Sebah, P.
"Constants and Records of Computation" . Retrieved 11 September 2007 .
^
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Further reading
Adamchik, Victor (2002).
"A certain series associated with Catalan's constant" . Zeitschrift für Analysis und ihre Anwendungen . 21 (3): 1–10.
doi :
10.4171/ZAA/1110 .
MR
1929434 . Archived from
the original on 2010-03-16. Retrieved 2005-07-14 .
Fee, Gregory J. (1990). "Computation of Catalan's Constant Using Ramanujan's Formula". In Watanabe, Shunro; Nagata, Morio (eds.). Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC '90, Tokyo, Japan, August 20-24, 1990 . ACM. pp. 157–160.
doi :
10.1145/96877.96917 .
ISBN
0201548925 .
S2CID
1949187 .
Bradley, David M. (1999). "A class of series acceleration formulae for Catalan's constant". The Ramanujan Journal . 3 (2): 159–173.
arXiv :
0706.0356 .
Bibcode :
2007arXiv0706.0356B .
doi :
10.1023/A:1006945407723 .
MR
1703281 .
S2CID
5111792 .
External links
Adamchik, Victor.
"33 representations for Catalan's constant" . Archived from
the original on 2016-08-07. Retrieved 14 July 2005 .
Plouffe, Simon (1993).
"A few identities (III) with Catalan" . Archived from
the original on 2019-06-26. Retrieved 29 July 2005 . (Provides over one hundred different identities).
Plouffe, Simon (1999).
"A few identities with Catalan constant and Pi^2" . Archived from
the original on 2019-06-26. Retrieved 29 July 2005 . (Provides a graphical interpretation of the relations)
Fee, Greg (1996).
Catalan's Constant (Ramanujan's Formula) . (Provides the first 300,000 digits of Catalan's constant)
Bradley, David M. (2001).
Representations of Catalan's constant .
CiteSeerX
10.1.1.26.1879 .
Johansson, Fredrik.
"0.915965594177219015054603514932" . Ordner, a catalog of real numbers in Fungrim . Retrieved 21 April 2021 .
"Catalan's Constant" .
YouTube . Let's Learn, Nemo!. 10 August 2020. Retrieved 6 April 2021 .
Weisstein, Eric W.
"Catalan's Constant" .
MathWorld .
"Catalan constant: Series representations" .
Wolfram Functions Site.
"Catalan constant" .
Encyclopedia of Mathematics .
EMS Press . 2001 [1994].