Lemniscate constant

Ratio of the perimeter of Bernoulli's lemniscate to its diameter
Lemniscate of Bernoulli

In mathematics, the lemniscate constant ϖ[1][2][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

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 {\tfrac {1}{M\left(1,{\sqrt {2}}\right)}}} .[6] By 1799, Gauss had two proofs of the theorem that M ( 1 , 2 ) = π ϖ {\displaystyle M\left(1,{\sqrt {2}}\right)={\tfrac {\pi }{\varpi }}} where ϖ {\displaystyle \varpi } is the lemniscate constant.[2][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 \left\{\pi ,M\left(1,{\tfrac {1}{\sqrt {2}}}\right),M'\left(1,{\tfrac {1}{\sqrt {2}}}\right)\right\}} (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}\left(1,{\frac {1}{\sqrt {2}}}\right)}{M'\left(1,{\frac {1}{\sqrt {2}}}\right)}}={\frac {1}{G^{3}M'\left(1,{\frac {1}{\sqrt {2}}}\right)}}.}

Forms

Usually, ϖ {\displaystyle \varpi } is defined by the first equality below.[2][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 }\left({\sqrt[{4}]{1+t^{4}}}-t\right)\,\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} \left({\tfrac {1}{4}},{\tfrac {1}{2}}\right)={\frac {\Gamma \left({\frac {1}{4}}\right)^{2}}{2{\sqrt {2\pi }}}}={\frac {2-{\sqrt {2}}}{4}}{\frac {\zeta \left({\frac {3}{4}}\right)^{2}}{\zeta \left({\frac {1}{4}}\right)^{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\left(1,{\sqrt {2}}\right)}}.}

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\left(1,{\sqrt {2}}\right)} published in 1800:[24]

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} \left({\tfrac {1}{4}},{\tfrac {1}{2}}\right)}

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 \left({\tfrac {1}{4}}\right){}^{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} \left({\tfrac {1}{4}},{\tfrac {1}{2}}\right),\\[3mu]B&={\frac {1}{2G}}={\tfrac {1}{4}}\mathrm {B} \left({\tfrac {1}{2}},{\tfrac {3}{4}}\right).\end{aligned}}}

Series

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 {\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {\frac {1}{2}}{\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)=\left({\frac {2}{1}}\cdot {\frac {2}{3}}\right)\left({\frac {4}{3}}\cdot {\frac {4}{5}}\right)\left({\frac {6}{5}}\cdot {\frac {6}{7}}\right)\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)=\left({\frac {3}{2}}\cdot {\frac {4}{5}}\right)\left({\frac {7}{6}}\cdot {\frac {8}{9}}\right)\left({\frac {11}{10}}\cdot {\frac {12}{13}}\right)\cdots }

A related result for Gauss's constant ( G = ϖ π {\displaystyle G={\tfrac {\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)=\left({\frac {3}{4}}\cdot {\frac {6}{5}}\right)\left({\frac {7}{8}}\cdot {\frac {10}{9}}\right)\left({\frac {11}{12}}\cdot {\frac {14}{13}}\right)\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 = 1 2 π ( ∑ n ∈ Z e − π n 2 ) 2 = 2 4 π e − π 12 ( ∑ n ∈ Z ( − 1 ) n e − π p n ) 2 {\displaystyle \varpi ={\frac {1}{\sqrt {2}}}\pi \left(\sum _{n\in \mathbb {Z} }e^{-\pi n^{2}}\right)^{2}={\sqrt[{4}]{2}}\pi e^{-{\frac {\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}={\tfrac {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 (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

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]

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

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} \left({\tfrac {1}{n}},{\tfrac {1}{n}}\right)} , 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 . {\displaystyle 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

Gauss's constant satisfies the equation[41]

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][41]

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[43]

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}

Other limits

Analogously to

2 π = lim n → ∞ | ( 2 n ) ! B 2 n | 1 2 n {\displaystyle 2\pi =\lim _{n\to \infty }\left|{\frac {(2n)!}{\mathrm {B} _{2n}}}\right|^{\frac {1}{2n}}}
where B n {\displaystyle \mathrm {B} _{n}} are Bernoulli numbers, we have
2 ϖ = lim n → ∞ ( ( 4 n ) ! H 4 n ) 1 4 n {\displaystyle 2\varpi =\lim _{n\to \infty }\left({\frac {(4n)!}{\mathrm {H} _{4n}}}\right)^{\frac {1}{4n}}}
where H n {\displaystyle \mathrm {H} _{n}} are Hurwitz numbers.

Notes

  1. ^ although neither of these proofs was rigorous from the modern point of view.
  2. ^ 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

  1. ^ Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404
  2. ^ a b c Cox 1984, p. 281.
  3. ^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 199
  4. ^ 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
  5. ^ Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203
  6. ^ a b Finch, Steven R. (18 August 2003). Mathematical Constants. Cambridge University Press. p. 420. ISBN 978-0-521-81805-6.
  7. ^ 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
  8. ^ Asai, Tetsuya (2007), Elliptic Gauss Sums and Hecke L-values at s=1, arXiv:0707.3711
  9. ^ "A062539 - Oeis".
  10. ^ "A014549 - Oeis".
  11. ^ 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.
  12. ^ a b "A085565 - Oeis".
  13. ^ "A076390 - Oeis".
  14. ^ 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.
  15. ^ "A064853 - Oeis".
  16. ^ "Lemniscate Constant".
  17. ^ 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.
  18. ^ G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
  19. ^ G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6
  20. ^ 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
  21. ^ Finch, Steven R. (18 August 2003). Mathematical Constants. Cambridge University Press. pp. 420–422. ISBN 978-0-521-81805-6.
  22. ^ 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.
  23. ^ "A113847 - Oeis".
  24. ^ Cox 1984, p. 277.
  25. ^ Levin (2006)
  26. ^ 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.
  27. ^ 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.
  28. ^ 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
  29. ^ Todd (1975)
  30. ^ Cox 1984, p. 307, eq. 2.21 for the first equality. The second equality can be proved by using the pentagonal number theorem.
  31. ^ Berndt, Bruce C. (1998). Ramanujan's Notebooks Part V. Springer. ISBN 978-1-4612-7221-2. p. 326
  32. ^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 232
  33. ^ Garrett, Paul. "Level-one elliptic modular forms" (PDF). University of Minnesota. p. 11—13
  34. ^ 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}} .
  35. ^ Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1. p. 146, 155
  36. ^ Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
  37. ^ "A062540 - OEIS". oeis.org. Retrieved 2022-09-14.
  38. ^ "A053002 - OEIS". oeis.org.
  39. ^ 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.
  40. ^ "A068467 - Oeis".
  41. ^ a b Cox 1984, p. 313.
  42. ^ Levien (2008)
  43. ^ Cox 1984, p. 312.
  44. ^ 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 } .
  • Weisstein, Eric W. "Gauss's Constant". MathWorld.
  • Sequences A014549, A053002, and A062539 in OEIS
  • Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.

External links

  • "Gauss's constant and where it occurs". www.johndcook.com. 2021-10-17.