Ratio of the perimeter of Bernoulli's lemniscate to its diameter
Lemniscate of Bernoulli In mathematics , the lemniscate constant ϖ [1] [3] [4] [5] is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter , analogous to the definition of π for the circle. Equivalently, the perimeter of the lemniscate ( x 2 + y 2 ) 2 = x 2 − y 2 {\displaystyle (x^{2}+y^{2})^{2}=x^{2}-y^{2}} is 2ϖ . The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755.[6] [7] [8] [9] The symbol ϖ is a cursive variant of π ; see Pi § Variant pi .
Gauss's constant , denoted by G , is equal to ϖ /π ≈ 0.8346268 .[10]
John Todd named two more lemniscate constants, the first lemniscate constant A = ϖ /2 ≈ 1.3110287771 and the second lemniscate constant B = π /(2ϖ ) ≈ 0.5990701173 .[11] [12] [13] [14]
Sometimes the quantities 2ϖ or A are referred to as the lemniscate constant.[15] [16]
History [ edit ] Gauss's constant G {\displaystyle G} is named after Carl Friedrich Gauss , who calculated it via the arithmetic–geometric mean as 1 / M ( 1 , 2 ) {\displaystyle 1/M(1,{\sqrt {2}})} .[6] By 1799, Gauss had two proofs of the theorem that M ( 1 , 2 ) = π / ϖ {\displaystyle M(1,{\sqrt {2}})=\pi /\varpi } where ϖ {\displaystyle \varpi } is the lemniscate constant.[a]
The lemniscate constant ϖ {\displaystyle \varpi } and first lemniscate constant A {\displaystyle A} were proven transcendental by Theodor Schneider in 1937 and the second lemniscate constant B {\displaystyle B} and Gauss's constant G {\displaystyle G} were proven transcendental by Theodor Schneider in 1941.[11] [17] [b] In 1975, Gregory Chudnovsky proved that the set { π , ϖ } {\displaystyle \{\pi ,\varpi \}} is algebraically independent over Q {\displaystyle \mathbb {Q} } , which implies that A {\displaystyle A} and B {\displaystyle B} are algebraically independent as well.[18] [19] But the set { π , M ( 1 , 1 / 2 ) , M ′ ( 1 , 1 / 2 ) } {\displaystyle \{\pi ,M(1,1/{\sqrt {2}}),M'(1,1/{\sqrt {2}})\}} (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over Q {\displaystyle \mathbb {Q} } . In fact,[20]
π = 2 2 M 3 ( 1 , 1 / 2 ) M ′ ( 1 , 1 / 2 ) = 1 G 3 M ′ ( 1 , 1 / 2 ) . {\displaystyle \pi =2{\sqrt {2}}{\frac {M^{3}(1,1/{\sqrt {2}})}{M'(1,1/{\sqrt {2}})}}={\frac {1}{G^{3}M'(1,1/{\sqrt {2}})}}.} Usually, ϖ {\displaystyle \varpi } is defined by the first equality below.[21] [22]
ϖ = 2 ∫ 0 1 d t 1 − t 4 = 2 ∫ 0 ∞ d t 1 + t 4 = ∫ 0 1 d t t − t 3 = ∫ 1 ∞ d t t 3 − t = 4 ∫ 0 ∞ ( 1 + t 4 4 − t ) d t = 2 2 ∫ 0 1 1 − t 4 4 d t = 3 ∫ 0 1 1 − t 4 d t = 2 K ( i ) = 1 2 B ( 1 4 , 1 2 ) = Γ ( 1 / 4 ) 2 2 2 π = 2 − 2 4 ζ ( 3 / 4 ) 2 ζ ( 1 / 4 ) 2 = 2.62205 75542 92119 81046 48395 89891 11941 … , {\displaystyle {\begin{aligned}\varpi &=2\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}={\sqrt {2}}\int _{0}^{\infty }{\frac {\mathrm {d} t}{\sqrt {1+t^{4}}}}=\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {t-t^{3}}}}=\int _{1}^{\infty }{\frac {\mathrm {d} t}{\sqrt {t^{3}-t}}}\\[6mu]&=4\int _{0}^{\infty }{\Bigl (}{\sqrt[{4}]{1+t^{4}}}-t{\Bigr )}\,\mathrm {d} t=2{\sqrt {2}}\int _{0}^{1}{\sqrt[{4}]{1-t^{4}}}\mathop {\mathrm {d} t} =3\int _{0}^{1}{\sqrt {1-t^{4}}}\,\mathrm {d} t\\[2mu]&=2K(i)={\tfrac {1}{2}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}={\frac {\Gamma (1/4)^{2}}{2{\sqrt {2\pi }}}}={\frac {2-{\sqrt {2}}}{4}}{\frac {\zeta (3/4)^{2}}{\zeta (1/4)^{2}}}\\[5mu]&=2.62205\;75542\;92119\;81046\;48395\;89891\;11941\ldots ,\end{aligned}}} where K is the complete elliptic integral of the first kind with modulus k , Β is the beta function , Γ is the gamma function and ζ is the Riemann zeta function .
The lemniscate constant can also be computed by the arithmetic–geometric mean M {\displaystyle M} ,
ϖ = π M ( 1 , 2 ) . {\displaystyle \varpi ={\frac {\pi }{M(1,{\sqrt {2}})}}.} Moreover,
e β ′ ( 0 ) = ϖ π {\displaystyle e^{\beta '(0)}={\frac {\varpi }{\sqrt {\pi }}}} which is analogous to
e ζ ′ ( 0 ) = 1 2 π {\displaystyle e^{\zeta '(0)}={\frac {1}{\sqrt {2\pi }}}} where β {\displaystyle \beta } is the Dirichlet beta function and ζ {\displaystyle \zeta } is the Riemann zeta function .[23]
Gauss's constant is typically defined as the reciprocal of the arithmetic–geometric mean of 1 and the square root of 2 , after his calculation of M ( 1 , 2 ) {\displaystyle M(1,{\sqrt {2}})} published in 1800:
G = 1 M ( 1 , 2 ) {\displaystyle G={\frac {1}{M(1,{\sqrt {2}})}}} Gauss's constant is equal to
G = 1 2 π B ( 1 4 , 1 2 ) {\displaystyle G={\frac {1}{2\pi }}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}} where Β denotes the beta function . A formula for G in terms of Jacobi theta functions is given by
G = ϑ 01 2 ( e − π ) {\displaystyle G=\vartheta _{01}^{2}\left(e^{-\pi }\right)} Gauss's constant may be computed from the gamma function at argument 1 / 4 :
G = Γ ( 1 4 ) 2 2 2 π 3 {\displaystyle G={\frac {\Gamma {\bigl (}{\tfrac {1}{4}}{\bigr )}{}^{2}}{2{\sqrt {2\pi ^{3}}}}}} John Todd's lemniscate constants may be given in terms of the beta function B:
A = 1 2 π G = 1 2 ϖ = 1 4 B ( 1 4 , 1 2 ) , B = 1 2 G = 1 4 B ( 1 2 , 3 4 ) . {\displaystyle {\begin{aligned}A&={\tfrac {1}{2}}\pi G={\tfrac {1}{2}}\varpi ={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )},\\[3mu]B&={\frac {1}{2G}}={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {3}{4}}{\bigr )}.\end{aligned}}} Viète's formula for π can be written:
2 π = 1 2 ⋅ 1 2 + 1 2 1 2 ⋅ 1 2 + 1 2 1 2 + 1 2 1 2 ⋯ {\displaystyle {\frac {2}{\pi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}}}\cdots } An analogous formula for ϖ is:[25]
2 ϖ = 1 2 ⋅ 1 2 + 1 2 / 1 2 ⋅ 1 2 + 1 2 / 1 2 + 1 2 / 1 2 ⋯ {\displaystyle {\frac {2}{\varpi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\bigg /}\!{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\Bigg /}\!{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\bigg /}\!{\sqrt {\frac {1}{2}}}}}}}\cdots } The Wallis product for π is:
π 2 = ∏ n = 1 ∞ ( 1 + 1 n ) ( − 1 ) n + 1 = ∏ n = 1 ∞ ( 2 n 2 n − 1 ⋅ 2 n 2 n + 1 ) = ( 2 1 ⋅ 2 3 ) ( 4 3 ⋅ 4 5 ) ( 6 5 ⋅ 6 7 ) ⋯ {\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }\left(1+{\frac {1}{n}}\right)^{(-1)^{n+1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\biggl (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\biggr )}{\biggl (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\biggr )}{\biggl (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\biggr )}\cdots } An analogous formula for ϖ is:[26]
ϖ 2 = ∏ n = 1 ∞ ( 1 + 1 2 n ) ( − 1 ) n + 1 = ∏ n = 1 ∞ ( 4 n − 1 4 n − 2 ⋅ 4 n 4 n + 1 ) = ( 3 2 ⋅ 4 5 ) ( 7 6 ⋅ 8 9 ) ( 11 10 ⋅ 12 13 ) ⋯ {\displaystyle {\frac {\varpi }{2}}=\prod _{n=1}^{\infty }\left(1+{\frac {1}{2n}}\right)^{(-1)^{n+1}}=\prod _{n=1}^{\infty }\left({\frac {4n-1}{4n-2}}\cdot {\frac {4n}{4n+1}}\right)={\biggl (}{\frac {3}{2}}\cdot {\frac {4}{5}}{\biggr )}{\biggl (}{\frac {7}{6}}\cdot {\frac {8}{9}}{\biggr )}{\biggl (}{\frac {11}{10}}\cdot {\frac {12}{13}}{\biggr )}\cdots } A related result for Gauss's constant ( G = ϖ / π {\displaystyle G=\varpi /\pi } ) is:[27]
G = ∏ n = 1 ∞ ( 4 n − 1 4 n ⋅ 4 n + 2 4 n + 1 ) = ( 3 4 ⋅ 6 5 ) ( 7 8 ⋅ 10 9 ) ( 11 12 ⋅ 14 13 ) ⋯ {\displaystyle G=\prod _{n=1}^{\infty }\left({\frac {4n-1}{4n}}\cdot {\frac {4n+2}{4n+1}}\right)={\biggl (}{\frac {3}{4}}\cdot {\frac {6}{5}}{\biggr )}{\biggl (}{\frac {7}{8}}\cdot {\frac {10}{9}}{\biggr )}{\biggl (}{\frac {11}{12}}\cdot {\frac {14}{13}}{\biggr )}\cdots } An infinite series of Gauss's constant discovered by Gauss is:[28]
G = ∑ n = 0 ∞ ( − 1 ) n ∏ k = 1 n ( 2 k − 1 ) 2 ( 2 k ) 2 = 1 − 1 2 2 2 + 1 2 ⋅ 3 2 2 2 ⋅ 4 2 − 1 2 ⋅ 3 2 ⋅ 5 2 2 2 ⋅ 4 2 ⋅ 6 2 + ⋯ {\displaystyle G=\sum _{n=0}^{\infty }(-1)^{n}\prod _{k=1}^{n}{\frac {(2k-1)^{2}}{(2k)^{2}}}=1-{\frac {1^{2}}{2^{2}}}+{\frac {1^{2}\cdot 3^{2}}{2^{2}\cdot 4^{2}}}-{\frac {1^{2}\cdot 3^{2}\cdot 5^{2}}{2^{2}\cdot 4^{2}\cdot 6^{2}}}+\cdots } The Machin formula for π is 1 4 π = 4 arctan 1 5 − arctan 1 239 , {\textstyle {\tfrac {1}{4}}\pi =4\arctan {\tfrac {1}{5}}-\arctan {\tfrac {1}{239}},} and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula 1 4 π = arctan 1 2 + arctan 1 3 {\textstyle {\tfrac {1}{4}}\pi =\arctan {\tfrac {1}{2}}+\arctan {\tfrac {1}{3}}} . Analogous formulas can be developed for ϖ , including the following found by Gauss: 1 2 ϖ = 2 arcsl 1 2 + arcsl 7 23 {\displaystyle {\tfrac {1}{2}}\varpi =2\operatorname {arcsl} {\tfrac {1}{2}}+\operatorname {arcsl} {\tfrac {7}{23}}} , where arcsl {\displaystyle \operatorname {arcsl} } is the lemniscate arcsine .[29]
The lemniscate constant can be rapidly computed by the series[30] [31]
ϖ = 2 − 1 / 2 π ( ∑ n ∈ Z e − π n 2 ) 2 = 2 1 / 4 π e − π / 12 ( ∑ n ∈ Z ( − 1 ) n e − π p n ) 2 {\displaystyle \varpi =2^{-1/2}\pi \left(\sum _{n\in \mathbb {Z} }e^{-\pi n^{2}}\right)^{2}=2^{1/4}\pi e^{-\pi /12}\left(\sum _{n\in \mathbb {Z} }(-1)^{n}e^{-\pi p_{n}}\right)^{2}} where p n = ( 3 n 2 − n ) / 2 {\displaystyle p_{n}=(3n^{2}-n)/2} (these are the generalized pentagonal numbers ).
In a spirit similar to that of the Basel problem ,
∑ z ∈ Z [ i ] ∖ { 0 } 1 z 4 = G 4 ( i ) = ϖ 4 15 {\displaystyle \sum _{z\in \mathbb {Z} [i]\setminus \{0\}}{\frac {1}{z^{4}}}=G_{4}(i)={\frac {\varpi ^{4}}{15}}} where Z [ i ] {\displaystyle \mathbb {Z} [i]} are the Gaussian integers and G 4 {\displaystyle G_{4}} is the Eisenstein series of weight 4 {\displaystyle 4} (see Lemniscate elliptic functions § Hurwitz numbers for a more general result).[32]
A related result is
∑ n = 1 ∞ σ 3 ( n ) e − 2 π n = ϖ 4 80 π 4 − 1 240 {\displaystyle \sum _{n=1}^{\infty }\sigma _{3}(n)e^{-2\pi n}={\frac {\varpi ^{4}}{80\pi ^{4}}}-{\frac {1}{240}}} where σ 3 {\displaystyle \sigma _{3}} is the sum of positive divisors function .[33]
In 1842, Malmsten found
∑ n = 1 ∞ ( − 1 ) n + 1 log ( 2 n + 1 ) 2 n + 1 = π 4 ( γ + 2 log π ϖ 2 ) {\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}{\frac {\log(2n+1)}{2n+1}}={\frac {\pi }{4}}\left(\gamma +2\log {\frac {\pi }{\varpi {\sqrt {2}}}}\right)} where γ {\displaystyle \gamma } is Euler's constant .
Gauss's constant is given by the rapidly converging series
G = 32 4 e − π 3 ( ∑ n = − ∞ ∞ ( − 1 ) n e − 2 n π ( 3 n + 1 ) ) 2 . {\displaystyle G={\sqrt[{4}]{32}}e^{-{\frac {\pi }{3}}}\left(\sum _{n=-\infty }^{\infty }(-1)^{n}e^{-2n\pi (3n+1)}\right)^{2}.} The constant is also given by the infinite product
G = ∏ m = 1 ∞ tanh 2 ( π m 2 ) . {\displaystyle G=\prod _{m=1}^{\infty }\tanh ^{2}\left({\frac {\pi m}{2}}\right).} Continued fractions [ edit ] A (generalized) continued fraction for π is
π 2 = 1 + 1 1 + 1 ⋅ 2 1 + 2 ⋅ 3 1 + 3 ⋅ 4 1 + ⋱ {\displaystyle {\frac {\pi }{2}}=1+{\cfrac {1}{1+{\cfrac {1\cdot 2}{1+{\cfrac {2\cdot 3}{1+{\cfrac {3\cdot 4}{1+\ddots }}}}}}}}} An analogous formula for
ϖ is
[12] ϖ 2 = 1 + 1 2 + 2 ⋅ 3 2 + 4 ⋅ 5 2 + 6 ⋅ 7 2 + ⋱ {\displaystyle {\frac {\varpi }{2}}=1+{\cfrac {1}{2+{\cfrac {2\cdot 3}{2+{\cfrac {4\cdot 5}{2+{\cfrac {6\cdot 7}{2+\ddots }}}}}}}}} Define Brouncker 's continued fraction by[34]
b ( s ) = s + 1 2 2 s + 3 2 2 s + 5 2 2 s + ⋱ , s > 0. {\displaystyle b(s)=s+{\cfrac {1^{2}}{2s+{\cfrac {3^{2}}{2s+{\cfrac {5^{2}}{2s+\ddots }}}}}},\quad s>0.} Let
n ≥ 0 {\displaystyle n\geq 0} except for the first equality where
n ≥ 1 {\displaystyle n\geq 1} . Then
[35] [36] b ( 4 n ) = ( 4 n + 1 ) ∏ k = 1 n ( 4 k − 1 ) 2 ( 4 k − 3 ) ( 4 k + 1 ) π ϖ 2 b ( 4 n + 1 ) = ( 2 n + 1 ) ∏ k = 1 n ( 2 k ) 2 ( 2 k − 1 ) ( 2 k + 1 ) 4 π b ( 4 n + 2 ) = ( 4 n + 1 ) ∏ k = 1 n ( 4 k − 3 ) ( 4 k + 1 ) ( 4 k − 1 ) 2 ϖ 2 π b ( 4 n + 3 ) = ( 2 n + 1 ) ∏ k = 1 n ( 2 k − 1 ) ( 2 k + 1 ) ( 2 k ) 2 π . {\displaystyle {\begin{aligned}b(4n)&=(4n+1)\prod _{k=1}^{n}{\frac {(4k-1)^{2}}{(4k-3)(4k+1)}}{\frac {\pi }{\varpi ^{2}}}\\b(4n+1)&=(2n+1)\prod _{k=1}^{n}{\frac {(2k)^{2}}{(2k-1)(2k+1)}}{\frac {4}{\pi }}\\b(4n+2)&=(4n+1)\prod _{k=1}^{n}{\frac {(4k-3)(4k+1)}{(4k-1)^{2}}}{\frac {\varpi ^{2}}{\pi }}\\b(4n+3)&=(2n+1)\prod _{k=1}^{n}{\frac {(2k-1)(2k+1)}{(2k)^{2}}}\,\pi .\end{aligned}}} For example,
b ( 1 ) = 4 π b ( 2 ) = ϖ 2 π b ( 3 ) = π b ( 4 ) = 9 π ϖ 2 . {\displaystyle {\begin{aligned}b(1)&={\frac {4}{\pi }}\\b(2)&={\frac {\varpi ^{2}}{\pi }}\\b(3)&=\pi \\b(4)&={\frac {9\pi }{\varpi ^{2}}}.\end{aligned}}} Simple continued fractions[37] [38] [ edit ]
ϖ = [ 2 , 1 , 1 , 1 , 1 , 1 , 4 , 1 , 2 , … ] 2 ϖ = [ 5 , 4 , 10 , 2 , 1 , 2 , 3 , 29 , … ] ϖ 2 = [ 1 , 3 , 4 , 1 , 1 , 1 , 5 , 2 , … ] G = [ 0 , 1 , 5 , 21 , 3 , 4 , 14 , … ] {\displaystyle {\begin{aligned}\varpi &=[2,1,1,1,1,1,4,1,2,\ldots ]\\2\varpi &=[5,4,10,2,1,2,3,29,\ldots ]\\{\frac {\varpi }{2}}&=[1,3,4,1,1,1,5,2,\ldots ]\\G&=[0,1,5,21,3,4,14,\ldots ]\end{aligned}}} Integrals [ edit ] A geometric representation of ϖ / 2 {\displaystyle \varpi /2} and ϖ / 2 {\displaystyle \varpi /{\sqrt {2}}} ϖ is related to the area under the curve x 4 + y 4 = 1 {\displaystyle x^{4}+y^{4}=1} . Defining π n := B ( 1 n , 1 n ) {\displaystyle \pi _{n}\mathrel {:=} \mathrm {B} {\bigl (}{\tfrac {1}{n}},{\tfrac {1}{n}}{\bigr )}} , twice the area in the positive quadrant under the curve x n + y n = 1 {\displaystyle x^{n}+y^{n}=1} is 2 ∫ 0 1 1 − x n n d x = 1 n π n . {\textstyle 2\int _{0}^{1}{\sqrt[{n}]{1-x^{n}}}\mathop {\mathrm {d} x} ={\tfrac {1}{n}}\pi _{n}.} In the quartic case, 1 4 π 4 = 1 2 ϖ . {\displaystyle {\tfrac {1}{4}}\pi _{4}={\tfrac {1}{\sqrt {2}}}\varpi .}
In 1842, Malmsten discovered that[39]
∫ 0 1 log ( − log x ) 1 + x 2 d x = π 2 log π ϖ 2 . {\displaystyle \int _{0}^{1}{\frac {\log(-\log x)}{1+x^{2}}}\,dx={\frac {\pi }{2}}\log {\frac {\pi }{\varpi {\sqrt {2}}}}.} Furthermore,
∫ 0 ∞ tanh x x e − x d x = log ϖ 2 π {\displaystyle \int _{0}^{\infty }{\frac {\tanh x}{x}}e^{-x}\,dx=\log {\frac {\varpi ^{2}}{\pi }}} and[40]
∫ 0 ∞ e − x 4 d x = 2 ϖ 2 π 4 , analogous to ∫ 0 ∞ e − x 2 d x = π 2 , {\displaystyle \int _{0}^{\infty }e^{-x^{4}}\,dx={\frac {\sqrt {2\varpi {\sqrt {2\pi }}}}{4}},\quad {\text{analogous to}}\,\int _{0}^{\infty }e^{-x^{2}}\,dx={\frac {\sqrt {\pi }}{2}},} a form of
Gaussian integral .
Gauss's constant appears in the evaluation of the integrals
1 G = ∫ 0 π 2 sin ( x ) d x = ∫ 0 π 2 cos ( x ) d x {\displaystyle {\frac {1}{G}}=\int _{0}^{\frac {\pi }{2}}{\sqrt {\sin(x)}}\,dx=\int _{0}^{\frac {\pi }{2}}{\sqrt {\cos(x)}}\,dx}
G = ∫ 0 ∞ d x cosh ( π x ) {\displaystyle G=\int _{0}^{\infty }{\frac {dx}{\sqrt {\cosh(\pi x)}}}} The first and second lemniscate constants are defined by integrals:[11]
A = ∫ 0 1 d x 1 − x 4 {\displaystyle A=\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}}
B = ∫ 0 1 x 2 d x 1 − x 4 {\displaystyle B=\int _{0}^{1}{\frac {x^{2}\,dx}{\sqrt {1-x^{4}}}}} Circumference of an ellipse [ edit ] Gauss's constant satisfies the equation
1 G = 2 ∫ 0 1 x 2 d x 1 − x 4 {\displaystyle {\frac {1}{G}}=2\int _{0}^{1}{\frac {x^{2}\,dx}{\sqrt {1-x^{4}}}}} Euler discovered in 1738 that for the rectangular elastica (first and second lemniscate constants)[42]
arc length ⋅ height = A ⋅ B = ∫ 0 1 d x 1 − x 4 ⋅ ∫ 0 1 x 2 d x 1 − x 4 = ϖ 2 ⋅ π 2 ϖ = π 4 {\displaystyle {\textrm {arc}}\ {\textrm {length}}\cdot {\textrm {height}}=A\cdot B=\int _{0}^{1}{\frac {\mathrm {d} x}{\sqrt {1-x^{4}}}}\cdot \int _{0}^{1}{\frac {x^{2}\mathop {\mathrm {d} x} }{\sqrt {1-x^{4}}}}={\frac {\varpi }{2}}\cdot {\frac {\pi }{2\varpi }}={\frac {\pi }{4}}} Now considering the circumference C {\displaystyle C} of the ellipse with axes 2 {\displaystyle {\sqrt {2}}} and 1 {\displaystyle 1} , satisfying 2 x 2 + 4 y 2 = 1 {\displaystyle 2x^{2}+4y^{2}=1} , Stirling noted that
C 2 = ∫ 0 1 d x 1 − x 4 + ∫ 0 1 x 2 d x 1 − x 4 {\displaystyle {\frac {C}{2}}=\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}+\int _{0}^{1}{\frac {x^{2}\,dx}{\sqrt {1-x^{4}}}}} Hence the full circumference is
C = 1 G + G π ≈ 3.820197789 … {\displaystyle C={\frac {1}{G}}+G\pi \approx 3.820197789\ldots } This is also the arc length of the sine curve on half a period:[44]
C = ∫ 0 π 1 + cos 2 ( x ) d x {\displaystyle C=\int _{0}^{\pi }{\sqrt {1+\cos ^{2}(x)}}\,dx} ^ although neither of these proofs was rigorous from the modern point of view. ^ In particular, he proved that the beta function B ( a , b ) {\displaystyle \mathrm {B} (a,b)} is transcendental for all a , b ∈ Q ∖ Z {\displaystyle a,b\in \mathbb {Q} \setminus \mathbb {Z} } such that a + b ∉ Z 0 − {\displaystyle a+b\notin \mathbb {Z} _{0}^{-}} . The fact that ϖ {\displaystyle \varpi } is transcendental follows from ϖ = 1 2 B ( 1 4 , 1 2 ) {\displaystyle \varpi ={\tfrac {1}{2}}\mathrm {B} \left({\tfrac {1}{4}},{\tfrac {1}{2}}\right)} and similarly for B and G from B ( 1 2 , 3 4 ) . {\displaystyle \mathrm {B} \left({\tfrac {1}{2}},{\tfrac {3}{4}}\right).} References [ edit ] ^ Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404 ^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi . American Mathematical Society. ISBN 0-8218-3246-8 . p. 199 ^ Bottazzini, Umberto ; Gray, Jeremy (2013). Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory . Springer. doi :10.1007/978-1-4614-5725-1 . ISBN 978-1-4614-5724-4 . p. 57 ^ Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions . Springer. ISBN 978-4-431-54918-5 . p. 203 ^ a b Finch, Steven R. (18 August 2003). Mathematical Constants . Cambridge University Press. p. 420. ISBN 978-0-521-81805-6 . ^ Kobayashi, Hiroyuki; Takeuchi, Shingo (2019), "Applications of generalized trigonometric functions with two parameters", Communications on Pure & Applied Analysis , 18 (3): 1509–1521, arXiv :1903.07407 , doi :10.3934/cpaa.2019072 , S2CID 102487670 ^ Asai, Tetsuya (2007), Elliptic Gauss Sums and Hecke L-values at s=1 , arXiv :0707.3711 ^ "A062539 - Oeis" . ^ "A014549 - Oeis" . ^ a b c Todd, John (January 1975). "The lemniscate constants" . Communications of the ACM . 18 (1): 14–19. doi :10.1145/360569.360580 . S2CID 85873 . ^ a b "A085565 - Oeis" . ^ "A076390 - Oeis" . ^ Carlson, B. C. (2010), "Elliptic Integrals" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248 . ^ "A064853 - Oeis" . ^ "Lemniscate Constant" . ^ Schneider, Theodor (1941). "Zur Theorie der Abelschen Funktionen und Integrale" . Journal für die reine und angewandte Mathematik . 183 (19): 110–128. doi :10.1515/crll.1941.183.110 . S2CID 118624331 . ^ G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis , Notices of the AMS 22, 1975, p. A-486 ^ G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers , American Mathematical Society, 1984, p. 6 ^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7 . p. 45 ^ Finch, Steven R. (18 August 2003). Mathematical Constants . Cambridge University Press. pp. 420–422. ISBN 978-0-521-81805-6 . ^ Schappacher, Norbert (1997). "Some milestones of lemniscatomy" (PDF) . In Sertöz, S. (ed.). Algebraic Geometry (Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey). Marcel Dekker. pp. 257–290. ^ "A113847 - Oeis" . ^ Levin (2006) ^ Hyde (2014) proves the validity of a more general Wallis-like formula for clover curves; here the special case of the lemniscate is slightly transformed, for clarity. ^ Hyde, Trevor (2014). "A Wallis product on clovers" (PDF) . The American Mathematical Monthly . 121 (3): 237–243. doi :10.4169/amer.math.monthly.121.03.237 . S2CID 34819500 . ^ Bottazzini, Umberto ; Gray, Jeremy (2013). Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory . Springer. doi :10.1007/978-1-4614-5725-1 . ISBN 978-1-4614-5724-4 . p. 60 ^ Todd (1975) ^ Cox 1984 , p. 307, eq. 2.21 for the first equality. The second equality can be proved by using the pentagonal number theorem . ^ Berndt, Bruce C. (1998). Ramanujan's Notebooks Part V . Springer. ISBN 978-1-4612-7221-2 . p. 326 ^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi . American Mathematical Society. ISBN 0-8218-3246-8 . p. 232 ^ Garrett, Paul. "Level-one elliptic modular forms" (PDF) . University of Minnesota . p. 11—13 ^ Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1 . p. 140 (eq. 3.34), p. 153. There's an error on p. 153: 4 [ Γ ( 3 + s / 4 ) / Γ ( 1 + s / 4 ) ] 2 {\displaystyle 4[\Gamma (3+s/4)/\Gamma (1+s/4)]^{2}} should be 4 [ Γ ( ( 3 + s ) / 4 ) / Γ ( ( 1 + s ) / 4 ) ] 2 {\displaystyle 4[\Gamma ((3+s)/4)/\Gamma ((1+s)/4)]^{2}} . ^ Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1 . p. 146, 155 ^ Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24 ^ "A062540 - OEIS" . oeis.org . Retrieved 2022-09-14 . ^ "A053002 - OEIS" . oeis.org . ^ Blagouchine, Iaroslav V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results" . The Ramanujan Journal . 35 (1): 21–110. doi :10.1007/s11139-013-9528-5 . S2CID 120943474 . ^ "A068467 - Oeis" . ^ Levien (2008) ^ Adlaj, Semjon (2012). "An Eloquent Formula for the Perimeter of an Ellipse" (PDF) . American Mathematical Society . p. 1097. One might also observe that the length of the "sine" curve over half a period, that is, the length of the graph of the function sin(t) from the point where t = 0 to the point where t = π , is 2 l ( 1 / 2 ) = L + M {\displaystyle {\sqrt {2}}l(1/{\sqrt {2}})=L+M} . In this paper M = 1 / G = π / ϖ {\displaystyle M=1/G=\pi /\varpi } and L = π / M = G π = ϖ {\displaystyle L=\pi /M=G\pi =\varpi } . External links [ edit ]