Lemniscate Constant

In 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, analogous to the definition of for the circle. Equivalently, the perimeter of the lemniscate $(x^2+y^2)^2=x^2-y^2$ is . The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755. The symbol is a cursive 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 $G$ is named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as $1/M(1,\sqrt)$. By 1799, Gauss had two proofs of the theorem that $M(1,\sqrt)=\pi/\varpi$ where $\varpi$ is the lemniscate constant.

The lemniscate constant $\varpi$ and first lemniscate constant $A$ were proven transcendental by Theodor Schneider in 1937 and the second lemniscate constant $B$ and Gauss's constant $G$ were proven transcendental by Theodor Schneider in 1941. In 1975, Gregory Chudnovsky proved that the set $\$ is algebraically independent over $\mathbb$, which implies that $A$ and $B$ are algebraically independent as well. But the set $\$ (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over $\mathbb$. In fact,

$\pi=2\sqrt\frac=\frac.$

Forms

Usually, $\varpi$ is defined by the first equality below.

\begin
\varpi
&= 2\int_0^1\frac
= \sqrt2\int_0^\infty\frac
= \int_0^1\frac
= \int_1^\infty \frac\\ mu&= 4\int_0^\infty\Bigl(\sqrt t\Bigr)\,\mathrmt
= 2\sqrt2\int_0^1 \sqrt mathop =3\int_0^1 \sqrt\,\mathrm dt\\ mu&= 2K(i)
= \tfrac\Beta\bigl( \tfrac14, \tfrac12\bigr)
= \frac
= \frac\frac\\ mu&= 2.62205\;75542\;92119\;81046\;48395\;89891\;11941\ldots,
\end

where is the complete elliptic integral of the first kind with modulus , is the beta function, is the gamma function and is the Riemann zeta function.

The lemniscate constant can also be computed by the arithmetic–geometric mean $M$,

$\varpi=\frac.$

Moreover,
$e^=\frac$

which is analogous to

$e^=\frac$

where $\beta$ is the Dirichlet beta function and $\zeta$ is the Riemann zeta function.

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 $M(1, \sqrt)$ published in 1800:

$G = \frac$

Gauss's constant is equal to

$G = \frac\Beta\bigl( \tfrac14, \tfrac12\bigr)$

where Β denotes the beta function. A formula for ''G'' in terms of Jacobi theta functions is given by

$G = \vartheta_^2\left(e^\right)$

Gauss's constant may be computed from the gamma function at argument :

$G = \frac$

John Todd's lemniscate constants may be given in terms of the beta function B:
\begin
A &= \tfrac12\pi G = \tfrac12\varpi = \tfrac14 \Beta \bigl(\tfrac14,\tfrac12\bigr), \\ muB &= \frac =\tfrac14\Beta \bigl(\tfrac12,\tfrac34\bigr).
\end

Series

Viète's formula for can be written:

$\frac2\pi = \sqrt\frac12 \cdot \sqrt \cdot \sqrt \cdots$

An analogous formula for is:

$\frac2\varpi = \sqrt\frac12 \cdot \sqrt \cdot \sqrt \cdots$

The Wallis product for is:

$\frac = \prod_^\infty \left(1+\frac\right)^=\prod_^ \left(\frac \cdot \frac\right) = \biggl(\frac \cdot \frac\biggr) \biggl(\frac \cdot \frac\biggr) \biggl(\frac \cdot \frac\biggr) \cdots$

An analogous formula for is:

$\frac = \prod_^\infty \left(1+\frac\right)^=\prod_^ \left(\frac \cdot \frac\right) = \biggl(\frac \cdot \frac\biggr) \biggl(\frac \cdot \frac\biggr) \biggl(\frac \cdot \frac\biggr) \cdots$

A related result for Gauss's constant ($G=\varpi / \pi$) is:

$G = \prod_^ \left(\frac \cdot \frac\right) = \biggl(\frac \cdot \frac\biggr) \biggl(\frac \cdot \frac\biggr) \biggl(\frac \cdot \frac\biggr) \cdots$

An infinite series of Gauss's constant discovered by Gauss is:

$G = \sum_^\infty (-1)^n \prod_^n \frac = 1 - \frac + \frac - \frac + \cdots$

The Machin formula for is $\tfrac14\pi = 4 \arctan \tfrac15 - \arctan \tfrac1,$ and several similar formulas for can be developed using trigonometric angle sum identities, e.g. Euler's formula $\tfrac14\pi = \arctan\tfrac12 + \arctan\tfrac13$. Analogous formulas can be developed for , including the following found by Gauss: $\tfrac12\varpi = 2 \operatorname \tfrac12 + \operatorname \tfrac7$, where $\operatorname$ is the lemniscate arcsine.

The lemniscate constant can be rapidly computed by the series

:$\varpi=2^\pi\left(\sum_e^\right)^2=2^\pi e^ \left(\sum_(-1)^n e^\right)^2$

where $p_n=(3n^2-n)/2$ (these are the generalized pentagonal numbers).

In a spirit similar to that of the Basel problem,
:$\sum_\frac=G_4(i)=\frac$
where \mathbb /math> are the Gaussian integers and $G_4$ is the Eisenstein series of weight $4$ (see Lemniscate elliptic functions § Hurwitz numbers for a more general result).

A related result is
:$\sum_^\infty \sigma_3(n)e^=\frac-\frac$
where $\sigma_3$ is the sum of positive divisors function.

In 1842, Malmsten found
:$\sum_^\infty (-1)^\frac=\frac\left(\gamma+2\log\frac\right)$

where $\gamma$ is Euler's constant.

Gauss's constant is given by the rapidly converging series

G = \sqrt ^\left (\sum_^\infty (-1)^n e^ \right )^2.

The constant is also given by the infinite product

:$G = \prod_^\infty \tanh^2 \left( \frac\right).$

Continued fractions

The simple continued fraction of is given by

$\varpi=2 + \cfrac$

A (generalized) continued fraction for is
$\frac\pi2=1 + \cfrac$
An analogous formula for is
$\frac\varpi2= 1 + \cfrac$

Define '' Brouncker's continued fraction'' by
$b(s)=s + \cfrac,\quad s>0.$
Let $n\ge 0$ except for the first equality where $n\ge 1$. Then
$\beginb(4n)&=(4n+1)\prod_^n \frac\frac\\ b(4n+1)&=(2n+1)\prod_^n \frac\frac\\ b(4n+2)&=(4n+1)\prod_^n \frac\frac\\ b(4n+3)&=(2n+1)\prod_^n \frac\,\pi.\end$
For example,
$\beginb(1)&=\frac\\ b(2)&=\frac\\ b(3)&=\pi\\ b(4)&=\frac.\end$

Gauss' constant as a (simple) continued fraction is , 1, 5, 21, 3, 4, 14, ...

Integrals

is related to the area under the curve $x^4 + y^4 = 1$. Defining $\pi_n \mathrel \Beta\bigl(\tfrac1n, \tfrac1n \bigr)$, twice the area in the positive quadrant under the curve $x^n + y^n = 1$ is 2 \int_0^1 \sqrt mathop = \tfrac1n \pi_n. In the quartic case, $\tfrac14 \pi_4 = \tfrac1\sqrt \varpi.$

In 1842, Malmsten discovered that

$\int_0^1 \frac\, dx=\frac\log\frac.$

Furthermore,
$\int_0^\infty \frace^\, dx=\log\frac$

and

$\int_0^\infty e^\, dx=\frac,\quad\text\,\int_0^\infty e^\, dx=\frac,$
a form of Gaussian integral.

Gauss's constant appears in the evaluation of the integrals

$= \int_0^\sqrt\,dx=\int_0^\sqrt\,dx$

$G = \int_0^$

The first and second lemniscate constants are defined by integrals:

$A = \int_0^1\frac$

$B = \int_0^1\frac$

Circumference of an ellipse

Gauss's constant satisfies the equation

$\frac = 2 \int_0^1\frac$

Euler discovered in 1738 that for the rectangular elastica (first and second lemniscate constants)Levien (2008)

$\textrm\ \textrm\cdot\textrm = A \cdot B = \int_0^1 \frac \cdot \int_0^1 \frac = \frac\varpi2 \cdot \frac\pi = \frac\pi4$

Now considering the circumference $C$ of the ellipse with axes $\sqrt$ and $1$, satisfying $2x^2 + 4y^2 = 1$, Stirling noted that

$\frac = \int_0^1\frac + \int_0^1\frac$

Hence the full circumference is

$C = \frac + G \pi \approx 3.820197789\ldots$

This is also the arc length of the sine curve on half a period: In this paper $M=1/G=\pi/\varpi$ and $L = \pi/M=G\pi=\varpi$.

$C = \int_0^\pi \sqrt\,dx$

Notes

References

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* Sequences A014549, A053002, an
A062539
in OEIS
*

External links

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{{Irrational number

Mathematical constants
Real transcendental numbers