Gauss's constant, denoted by G, is equal to ϖ/π ≈ 0.8346268.[10]
John Todd named two more lemniscate constants, the first lemniscate constantA = ϖ/2 ≈ 1.3110287771 and the second lemniscate constantB = π/(2ϖ) ≈ 0.5990701173.[11][12][13][14]
Sometimes the quantities 2ϖ or A are referred to as the lemniscate constant.[15][16]
The lemniscate constant and first lemniscate constant were proven
transcendental by
Theodor Schneider in 1937 and the second lemniscate constant and Gauss's constant were proven transcendental by Theodor Schneider in 1941.[11][17][b] In 1975,
Gregory Chudnovsky proved that the set is
algebraically independent over , which implies that and are algebraically independent as well.[18][19] But the set (where the prime denotes the
derivative with respect to the second variable) is not algebraically independent over . In fact,[20]
Forms
Usually, is defined by the first equality below.[2][21][22]
An infinite series of Gauss's constant discovered by Gauss is:[28]
The
Machin formula for π is and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula . Analogous formulas can be developed for ϖ, including the following found by Gauss: , where is the
lemniscate arcsine.[29]
The lemniscate constant can be rapidly computed by the series[30][31]
^although neither of these proofs was rigorous from the modern point of view.
^In particular, he proved that the
beta function is transcendental for all such that . The fact that is transcendental follows from and similarly for B and G from
References
^Gauss, C. F. (1866).
Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404
^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.
ISBN0-471-83138-7. p. 45
^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.
^Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press.
ISBN978-0-521-85419-1. p. 140 (eq. 3.34), p. 153. There's an error on p. 153: should be .
^Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press.
ISBN978-0-521-85419-1. p. 146, 155
^Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
^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 . In this paper and .
Gauss's constant, denoted by G, is equal to ϖ/π ≈ 0.8346268.[10]
John Todd named two more lemniscate constants, the first lemniscate constantA = ϖ/2 ≈ 1.3110287771 and the second lemniscate constantB = π/(2ϖ) ≈ 0.5990701173.[11][12][13][14]
Sometimes the quantities 2ϖ or A are referred to as the lemniscate constant.[15][16]
The lemniscate constant and first lemniscate constant were proven
transcendental by
Theodor Schneider in 1937 and the second lemniscate constant and Gauss's constant were proven transcendental by Theodor Schneider in 1941.[11][17][b] In 1975,
Gregory Chudnovsky proved that the set is
algebraically independent over , which implies that and are algebraically independent as well.[18][19] But the set (where the prime denotes the
derivative with respect to the second variable) is not algebraically independent over . In fact,[20]
Forms
Usually, is defined by the first equality below.[2][21][22]
An infinite series of Gauss's constant discovered by Gauss is:[28]
The
Machin formula for π is and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula . Analogous formulas can be developed for ϖ, including the following found by Gauss: , where is the
lemniscate arcsine.[29]
The lemniscate constant can be rapidly computed by the series[30][31]
^although neither of these proofs was rigorous from the modern point of view.
^In particular, he proved that the
beta function is transcendental for all such that . The fact that is transcendental follows from and similarly for B and G from
References
^Gauss, C. F. (1866).
Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404
^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.
ISBN0-471-83138-7. p. 45
^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.
^Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press.
ISBN978-0-521-85419-1. p. 140 (eq. 3.34), p. 153. There's an error on p. 153: should be .
^Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press.
ISBN978-0-521-85419-1. p. 146, 155
^Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
^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 . In this paper and .