![Lemniscate of Bernoulli](https://upload.wikimedia.org/wikipedia/commons/e/ef/Lemniscate_of_Bernoulli.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the lemniscate constant
[ p. 199] is a
transcendental mathematical constant that is the ratio of the
perimeter of
Bernoulli's lemniscate to its
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
, analogous to the definition of
for the circle. Equivalently, the perimeter of the lemniscate
is . The lemniscate constant is closely related to the
lemniscate elliptic functions and approximately equal to 2.62205755.
The symbol is a
cursive
Cursive (also known as script, among other names) is any style of penmanship in which characters are written joined in a flowing manner, generally for the purpose of making writing faster, in contrast to block letters. It varies in functionalit ...
variant of ; see
Pi § Variant pi.
Gauss's constant, denoted by ''G'', is equal to .
John Todd named two more lemniscate constants, the ''first lemniscate constant'' and the ''second lemniscate constant'' .
Sometimes the quantities or are referred to as ''the'' lemniscate constant.
History
Gauss's constant
is named after
Carl Friedrich Gauss, who calculated it via the
arithmetic–geometric mean as
.
[ By 1799, Gauss had two proofs of the theorem that where is the lemniscate constant.
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.] In 1975, Gregory Chudnovsky
David Volfovich Chudnovsky (born January 22, 1947 in Kyiv) and Gregory Volfovich Chudnovsky (born April 17, 1952 in Kyiv) are Ukrainian-born American mathematicians and engineers known for their world-record mathematical calculations and developing ...
proved that the set is algebraically independent
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K.
In particular, a one element set \ is algebraically ind ...
over , which implies that and are algebraically independent as well. But the set (where the prime denotes the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
with respect to the second variable) is not algebraically independent over . In fact,
Forms
Usually, is defined by the first equality below.
where is the complete elliptic integral of the first kind
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
with modulus , is the beta function, is the gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
and is the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
The lemniscate constant can also be computed by the arithmetic–geometric mean ,
Moreover,
which is analogous to
where is the Dirichlet beta function and is the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
Gauss's constant is typically defined as the reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of the arithmetic–geometric mean of 1 and the square root of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
, after his calculation of published in 1800:
Gauss's constant is equal to
where Β denotes the beta function. A formula for ''G'' in terms of Jacobi theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
s is given by
Gauss's constant may be computed from the gamma function at argument :
John Todd's lemniscate constants may be given in terms of the beta function B:
Series
Viète's formula for can be written:
An analogous formula for is:
The Wallis product
In mathematics, the Wallis product for , published in 1656 by John Wallis, states that
:\begin
\frac & = \prod_^ \frac = \prod_^ \left(\frac \cdot \frac\right) \\ pt& = \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\fr ...
for is:
An analogous formula for is:
A related result for Gauss's constant () is:
An infinite series of Gauss's constant discovered by Gauss is:
The Machin formula for is and several similar formulas for can be developed using trigonometric angle sum identities, e.g. Euler's formula . Analogous formulas can be developed for , including the following found by Gauss: , where is the lemniscate arcsine.
The lemniscate constant can be rapidly computed by the series
:
where (these are the generalized pentagonal numbers).
In a spirit similar to that of the Basel problem,
:
where