Lemniscate elliptic functions – Wikipedia

Posted on April 17, 2021 by lordneo

Mathematical functions

The lemniscate sine (red) and lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric sine

y = sin(πx / ϖ)

(pale dashed red).

In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.^[1]

The lemniscate sine and lemniscate cosine functions, usually written with the symbols $sl$ and $cl$ (sometimes the symbols $sinlem$ and $coslem$ or $sin lemn$ and $cos lemn$ are used instead),^[2] are analogous to the trigonometric functions sine and cosine. While the trigonometric sine relates the arc length to the chord length in a unit-diameter circle

{displaystyle x^{2}+y^{2}=x,}

${displaystyle x^{2}+y^{2}=x,}$ ^[3] the lemniscate sine relates the arc length to the chord length of a lemniscate

{displaystyle {bigl (}x^{2}+y^{2}{bigr )}{}^{2}=x^{2}-y^{2}.}

${displaystyle {bigl (}x^{2}+y^{2}{bigr )}{}^{2}=x^{2}-y^{2}.}$

The lemniscate functions have periods related to a number

{displaystyle varpi =}

${displaystyle varpi =}$ 2.622057… called the lemniscate constant, the ratio of a lemniscate’s perimeter to its diameter. This number is a quartic analog of the (quadratic)

{displaystyle pi =}

${displaystyle pi =}$ 3.141592…, ratio of perimeter to diameter of a circle.

As complex functions, $sl$ and $cl$ have a square period lattice (a multiple of the Gaussian integers) with fundamental periods

{displaystyle {(1+i)varpi ,(1-i)varpi },}

${displaystyle {(1+i)varpi ,(1-i)varpi },}$ ^[4] and are a special case of two Jacobi elliptic functions on that lattice,

{displaystyle operatorname {sl} z=operatorname {sn} (z;i),}

${displaystyle operatorname {sl} z=operatorname {sn} (z;i),}$

{displaystyle operatorname {cl} z=operatorname {cd} (z;i)}

${displaystyle operatorname {cl} z=operatorname {cd} (z;i)}$ .

Similarly, the hyperbolic lemniscate sine $slh$ and hyperbolic lemniscate cosine $clh$ have a square period lattice with fundamental periods

{displaystyle {bigl {}{sqrt {2}}varpi ,{sqrt {2}}varpi i{bigr }}.}

${displaystyle {bigl {}{sqrt {2}}varpi ,{sqrt {2}}varpi i{bigr }}.}$

The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function

{displaystyle wp (z;a,0)}

${displaystyle wp (z;a,0)}$ .

Table of Contents

Lemniscate sine and cosine functions[edit]

Definitions[edit]

The lemniscate functions $sl$ and $cl$ can be defined as the solution to the initial value problem:^[5]

{displaystyle {frac {mathrm {d} }{mathrm {d} z}}operatorname {sl} z={bigl (}1+operatorname {sl} ^{2}z{bigr )}operatorname {cl} z, {frac {mathrm {d} }{mathrm {d} z}}operatorname {cl} z=-{bigl (}1+operatorname {cl} ^{2}z{bigr )}operatorname {sl} z, operatorname {sl} 0=0, operatorname {cl} 0=1,}

or equivalently as the inverses of an elliptic integral, the Schwarz–Christoffel map from the complex unit disk to a square with corners

{displaystyle {big {}{tfrac {1}{2}}varpi ,{tfrac {1}{2}}varpi i,-{tfrac {1}{2}}varpi ,-{tfrac {1}{2}}varpi i{big }}colon }

${displaystyle {big {}{tfrac {1}{2}}varpi ,{tfrac {1}{2}}varpi i,-{tfrac {1}{2}}varpi ,-{tfrac {1}{2}}varpi i{big }}colon }$ ^[6]

{displaystyle z=int _{0}^{operatorname {sl} z}{frac {mathrm {d} t}{sqrt {1-t^{4}}}}=int _{operatorname {cl} z}^{1}{frac {mathrm {d} t}{sqrt {1-t^{4}}}}.}

Beyond that square, the functions can be analytically continued to the whole complex plane by a series of reflections.

By comparison, the circular sine and cosine can be defined as the solution to the initial value problem:

{displaystyle {frac {mathrm {d} }{mathrm {d} z}}sin z=cos z, {frac {mathrm {d} }{mathrm {d} z}}cos z=-sin z, sin 0=0, cos 0=1,}

or as inverses of a map from the upper half-plane to a half-infinite strip with real part between

{displaystyle -{tfrac {1}{2}}pi ,{tfrac {1}{2}}pi }

${displaystyle -{tfrac {1}{2}}pi ,{tfrac {1}{2}}pi }$ and positive imaginary part:

{displaystyle z=int _{0}^{sin z}{frac {mathrm {d} t}{sqrt {1-t^{2}}}}=int _{cos z}^{1}{frac {mathrm {d} t}{sqrt {1-t^{2}}}}.}

Arc length of Bernoulli’s lemniscate[edit]

The lemniscate sine and cosine relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.

The trigonometric sine and cosine analogously relate the arc length of an arc of a unit-diameter circle to the distance of one endpoint from the origin.

The lemniscate of Bernoulli with half-width $1$ is the locus of points in the plane such that the product of their distances from the two focal points

{displaystyle F_{1}={bigl (}{-{tfrac {1}{sqrt {2}}}},0{bigr )}}

${displaystyle F_{1}={bigl (}{-{tfrac {1}{sqrt {2}}}},0{bigr )}}$ and

{displaystyle F_{2}={bigl (}{tfrac {1}{sqrt {2}}},0{bigr )}}

${displaystyle F_{2}={bigl (}{tfrac {1}{sqrt {2}}},0{bigr )}}$ is the constant

{displaystyle {tfrac {1}{2}}}

${tfrac {1}{2}}$ . This is a quartic curve satisfying the polar equation

{displaystyle r^{2}=cos 2theta }

${displaystyle r^{2}=cos 2theta }$ or the Cartesian equation

{displaystyle {bigl (}x^{2}+y^{2}{bigr )}{}^{2}=x^{2}-y^{2}.}

${displaystyle {bigl (}x^{2}+y^{2}{bigr )}{}^{2}=x^{2}-y^{2}.}$

The points on the lemniscate at distance

{displaystyle r}

$r$ from the origin are the intersections of the circle

{displaystyle x^{2}+y^{2}=r^{2}}

$x^2+y^2=r^2$ and the hyperbola

{displaystyle x^{2}-y^{2}=r^{4}}

${displaystyle x^{2}-y^{2}=r^{4}}$ . The intersection in the positive quadrant has Cartesian coordinates:

{displaystyle {big (}x(r),y(r){big )}={biggl (}!{sqrt {{tfrac {1}{2}}r^{2}{bigl (}1+r^{2}{bigr )}}},,{sqrt {{tfrac {1}{2}}r^{2}{bigl (}1-r^{2}{bigr )}}},{biggr )}.}

Using this parametrization with

{displaystyle rin [0,1]}

${displaystyle rin [0,1]}$ for a quarter of the lemniscate, the arc length from the origin to a point

{displaystyle {big (}x(r),y(r){big )}}

${displaystyle {big (}x(r),y(r){big )}}$ is:^[7]

{displaystyle {begin{aligned}&int _{0}^{r}{sqrt {x'(t)^{2}+y'(t)^{2}}}mathop {mathrm {d} t} \&quad {}=int _{0}^{r}{sqrt {{frac {(1+2t^{2})^{2}}{2(1+t^{2})}}+{frac {(1-2t^{2})^{2}}{2(1-t^{2})}}}}mathop {mathrm {d} t} \[6mu]&quad {}=int _{0}^{r}{frac {mathrm {d} t}{sqrt {1-t^{4}}}}\[6mu]&quad {}=operatorname {arcsl} r.end{aligned}}}

Likewise, the arc length from

{displaystyle (1,0)}

$(1,0)$ to

{displaystyle {big (}x(r),y(r){big )}}

${displaystyle {big (}x(r),y(r){big )}}$ is:

{displaystyle {begin{aligned}&int _{r}^{1}{sqrt {x'(t)^{2}+y'(t)^{2}}}mathop {mathrm {d} t} \&quad {}=int _{r}^{1}{frac {mathrm {d} t}{sqrt {1-t^{4}}}}\[6mu]&quad {}=operatorname {arccl} r={tfrac {1}{2}}varpi -operatorname {arcsl} r.end{aligned}}}

Or in the inverse direction, the lemniscate sine and cosine functions give the distance from the origin as functions of arc length from the origin and the point

{displaystyle (1,0)}

$(1,0)$ , respectively.

Analogously, the circular sine and cosine functions relate the chord length to the arc length for the unit diameter circle with polar equation

{displaystyle r=cos theta }

${displaystyle r=cos theta }$ or Cartesian equation

{displaystyle x^{2}+y^{2}=x,}

${displaystyle x^{2}+y^{2}=x,}$ using the same argument above but with the parametrization:

{displaystyle {big (}x(r),y(r){big )}={biggl (}r^{2},,{sqrt {r^{2}{bigl (}1-r^{2}{bigr )}}},{biggr )}.}

Alternatively, just as the unit circle

{displaystyle x^{2}+y^{2}=1}

$x^{2}+y^{2}=1$ is parametrized in terms of the arc length

{displaystyle s}

$s$ from the point

{displaystyle (1,0)}

$(1,0)$ by

{displaystyle (x(s),y(s))=(cos s,sin s),}

the lemniscate is parametrized in terms of the arc length

{displaystyle s}

$s$ from the point

{displaystyle (1,0)}

$(1,0)$ by^[8]

{displaystyle (x(s),y(s))=left({frac {operatorname {cl} s}{sqrt {1+operatorname {sl} ^{2}s}}},{frac {operatorname {sl} soperatorname {cl} s}{sqrt {1+operatorname {sl} ^{2}s}}}right)=left({tilde {operatorname {cl} }},s,{tilde {operatorname {sl} }},sright).}

The lemniscate integral and lemniscate functions satisfy an argument duplication identity discovered by Fagnano in 1718:^[9]

{displaystyle int _{0}^{z}{frac {mathrm {d} t}{sqrt {1-t^{4}}}}=2int _{0}^{u}{frac {mathrm {d} t}{sqrt {1-t^{4}}}},quad {text{if }}z={frac {2u{sqrt {1-u^{4}}}}{1+u^{4}}}{text{ and }}0leq uleq {sqrt {{sqrt {2}}-1}}.}

A lemniscate divided into 15 sections of equal arclength (red curves). Because the prime factors of 15 (3 and 5) are both Fermat primes, this polygon (in black) is constructible using a straightedge and compass.

Later mathematicians generalized this result. Analogously to the constructible polygons in the circle, the lemniscate can be divided into $n$ sections of equal arc length using only straightedge and compass if and only if $n$ is of the form

{displaystyle n=2^{k}p_{1}p_{2}cdots p_{m}}

${displaystyle n=2^{k}p_{1}p_{2}cdots p_{m}}$ where $k$ is a non-negative integer and each $p i$ (if any) is a distinct Fermat prime.^[10] The “if” part of the theorem was proved by Niels Abel in 1827–1828, and the “only if” part was proved by Michael Rosen in 1981.^[11] Equivalently, the lemniscate can be divided into $n$ sections of equal arc length using only straightedge and compass if and only if

{displaystyle varphi (n)}

$varphi (n)$ is a power of two (where

{displaystyle varphi }

$varphi$ is Euler’s totient function). The lemniscate is not assumed to be already drawn; the theorem refers to constructing the division points only.

Let

{displaystyle r_{j}=operatorname {sl} {dfrac {2jvarpi }{n}}}

${displaystyle r_{j}=operatorname {sl} {dfrac {2jvarpi }{n}}}$ . Then the $n$ -division points for the lemniscate

{displaystyle (x^{2}+y^{2})^{2}=x^{2}-y^{2}}

$(x^{2}+y^{2})^{2}=x^{2}-y^{2}$ are the points

{displaystyle left(r_{j}{sqrt {{tfrac {1}{2}}{bigl (}1+r_{j}^{2}{bigr )}}}, (-1)^{leftlfloor 4j/nrightrfloor }{sqrt {{tfrac {1}{2}}r_{j}^{2}{bigl (}1-r_{j}^{2}{bigr )}}}right),quad jin {1,2,ldots ,n}}

where

{displaystyle lfloor cdot rfloor }

$lfloor cdot rfloor$ is the floor function. See below for some specific values of

{displaystyle operatorname {sl} {dfrac {2varpi }{n}}}

${displaystyle operatorname {sl} {dfrac {2varpi }{n}}}$ .

Arc length of rectangular elastica[edit]

The lemniscate sine relates the arc length to the x coordinate in the rectangular elastica.

The inverse lemniscate sine also describes the arc length $s$ relative to the $x$ coordinate of the rectangular elastica.^[12] This curve has $y$ coordinate and arc length:

{displaystyle y=int _{x}^{1}{frac {t^{2}mathop {mathrm {d} t} }{sqrt {1-t^{4}}}},quad s=operatorname {arcsl} x=int _{0}^{x}{frac {mathrm {d} t}{sqrt {1-t^{4}}}}}

The rectangular elastica solves a problem posed by Jacob Bernoulli, in 1691, to describe the shape of an idealized flexible rod fixed in a vertical orientation at the bottom end and pulled down by a weight from the far end until it has been bent horizontal. Bernoulli’s proposed solution established Euler–Bernoulli beam theory, further developed by Euler in the 18th century.

Elliptic characterization[edit]

The lemniscate elliptic functions and an ellipse

Let

{displaystyle C}

$C$ be a point on the ellipse

{displaystyle x^{2}+2y^{2}=1}

${displaystyle x^{2}+2y^{2}=1}$ in the first quadrant and let

{displaystyle D}

$D$ be the projection of

{displaystyle C}

$C$ on the unit circle

{displaystyle x^{2}+y^{2}=1}

$x^{2}+y^{2}=1$ . The distance

{displaystyle r}

$r$ between the origin

{displaystyle A}

$A$ and the point

{displaystyle C}

$C$ is a function of

{displaystyle varphi }

$varphi$ (the angle

{displaystyle BAC}

${displaystyle BAC}$ where

{displaystyle B=(1,0)}

${displaystyle B=(1,0)}$ ; equivalently the length of the circular arc

{displaystyle BD}

$BD$ ). The parameter

{displaystyle u}

$u$ is given by

{displaystyle u=int _{0}^{varphi }r(theta ),mathrm {d} theta =int _{0}^{varphi }{frac {mathrm {d} theta }{sqrt {1+sin ^{2}theta }}}.}

{displaystyle E}

$E$ is the projection of

{displaystyle D}

$D$ on the x-axis and if

{displaystyle F}

$F$ is the projection of

{displaystyle C}

$C$ on the x-axis, then the lemniscate elliptic functions are given by

{displaystyle operatorname {cl} u={overline {AF}},quad operatorname {sl} u={overline {DE}},}

{displaystyle {tilde {operatorname {cl} }},u={overline {AF}}{overline {AC}},quad {tilde {operatorname {sl} }},u={overline {AF}}{overline {FC}}.}

Relation to the lemniscate constant[edit]

The lemniscate sine function and hyperbolic lemniscate sine functions are defined as inverses of elliptic integrals. The complete integrals are related to the lemniscate constant

ϖ

The lemniscate functions have minimal real period $2 ϖ$ ^[13] and fundamental complex periods

{displaystyle (1+i)varpi }

${displaystyle (1+i)varpi }$ and

{displaystyle (1-i)varpi }

${displaystyle (1-i)varpi }$ for a constant $ϖ$ called the lemniscate constant,^[14]

{displaystyle varpi =2int _{0}^{1}{frac {mathrm {d} t}{sqrt {1-t^{4}}}}=2.62205ldots }

The lemniscate functions satisfy the basic relation

{displaystyle operatorname {cl} z={operatorname {sl} }{bigl (}{tfrac {1}{2}}varpi -z{bigr )},}

${displaystyle operatorname {cl} z={operatorname {sl} }{bigl (}{tfrac {1}{2}}varpi -z{bigr )},}$ analogous to the relation

{displaystyle cos z={sin }{bigl (}{tfrac {1}{2}}pi -z{bigr )}.}

${displaystyle cos z={sin }{bigl (}{tfrac {1}{2}}pi -z{bigr )}.}$

The lemniscate constant $ϖ$ is a close analog of the circle constant $π$ , and many identities involving $π$ have analogues involving $ϖ$ , as identities involving the trigonometric functions have analogues involving the lemniscate functions. For example, Viète’s formula for $π$ can be written:

{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:^[15]

{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 Machin formula for $π$ is

{textstyle {tfrac {1}{4}}pi =4arctan {tfrac {1}{5}}-arctan {tfrac {1}{239}},}

${textstyle {tfrac {1}{4}}pi =4arctan {tfrac {1}{5}}-arctan {tfrac {1}{239}},}$ and several similar formulas for $π$ can be developed using trigonometric angle sum identities, e.g. Euler’s formula

{textstyle {tfrac {1}{4}}pi =arctan {tfrac {1}{2}}+arctan {tfrac {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:

{displaystyle {tfrac {1}{2}}varpi =2operatorname {arcsl} {tfrac {1}{2}}+operatorname {arcsl} {tfrac {7}{23}}.}

${displaystyle {tfrac {1}{2}}varpi =2operatorname {arcsl} {tfrac {1}{2}}+operatorname {arcsl} {tfrac {7}{23}}.}$ ^[16]

The lemniscate and circle constants were found by Gauss to be related to each-other by the arithmetic-geometric mean $M$ :^[17]

{displaystyle {frac {pi }{varpi }}=M{left(1,{sqrt {2}}!~right)}}

Zeros, poles and symmetries[edit]

{displaystyle operatorname {sl} }

The lemniscate functions $cl$ and $sl$ are even and odd functions, respectively,

{displaystyle {begin{aligned}operatorname {cl} (-z)&=operatorname {cl} z\[6mu]operatorname {sl} (-z)&=-operatorname {sl} zend{aligned}}}

At translations of

{displaystyle {tfrac {1}{2}}varpi ,}

${displaystyle {tfrac {1}{2}}varpi ,}$ $cl$ and $sl$ are exchanged, and at translations of

{displaystyle {tfrac {1}{2}}ivarpi }

${displaystyle {tfrac {1}{2}}ivarpi }$ they are additionally rotated and reciprocated:^[19]

{displaystyle {begin{aligned}{operatorname {cl} }{bigl (}zpm {tfrac {1}{2}}varpi {bigr )}&=mp operatorname {sl} z,&{operatorname {cl} }{bigl (}zpm {tfrac {1}{2}}ivarpi {bigr )}&={frac {mp i}{operatorname {sl} z}}\[6mu]{operatorname {sl} }{bigl (}zpm {tfrac {1}{2}}varpi {bigr )}&=pm operatorname {cl} z,&{operatorname {sl} }{bigl (}zpm {tfrac {1}{2}}ivarpi {bigr )}&={frac {pm i}{operatorname {cl} z}}end{aligned}}}

Doubling these to translations by a unit-Gaussian-integer multiple of

{displaystyle varpi }

$varpi$ (that is,

{displaystyle pm varpi }

${displaystyle pm varpi }$ or

{displaystyle pm ivarpi }

${displaystyle pm ivarpi }$ ), negates each function, an involution:

{displaystyle {begin{aligned}operatorname {cl} (z+varpi )&=operatorname {cl} (z+ivarpi )=-operatorname {cl} z\[4mu]operatorname {sl} (z+varpi )&=operatorname {sl} (z+ivarpi )=-operatorname {sl} zend{aligned}}}

As a result, both functions are invariant under translation by an even-Gaussian-integer multiple of

{displaystyle varpi }

$varpi$ .^[20] That is, a displacement

{displaystyle (a+bi)varpi ,}

${displaystyle (a+bi)varpi ,}$ with

{displaystyle a+b=2k}

${displaystyle a+b=2k}$ for integers $a$ , $b$ , and $k$ .

{displaystyle {begin{aligned}{operatorname {cl} }{bigl (}z+(1+i)varpi {bigr )}&={operatorname {cl} }{bigl (}z+(1-i)varpi {bigr )}=operatorname {cl} z\[4mu]{operatorname {sl} }{bigl (}z+(1+i)varpi {bigr )}&={operatorname {sl} }{bigl (}z+(1-i)varpi {bigr )}=operatorname {sl} zend{aligned}}}

This makes them elliptic functions (doubly periodic meromorphic functions in the complex plane) with a diagonal square period lattice of fundamental periods

{displaystyle (1+i)varpi }

${displaystyle (1+i)varpi }$ and

{displaystyle (1-i)varpi }

${displaystyle (1-i)varpi }$ .^[21] Elliptic functions with a square period lattice are more symmetrical than arbitrary elliptic functions, following the symmetries of the square.

Reflections and quarter-turn rotations of lemniscate function arguments have simple expressions:

{displaystyle {begin{aligned}operatorname {cl} {bar {z}}&={overline {operatorname {cl} z}}\[6mu]operatorname {sl} {bar {z}}&={overline {operatorname {sl} z}}\[4mu]operatorname {cl} iz&={frac {1}{operatorname {cl} z}}\[6mu]operatorname {sl} iz&=ioperatorname {sl} zend{aligned}}}

The $sl$ function has simple zeros at Gaussian integer multiples of $ϖ$ , complex numbers of the form

{displaystyle avarpi +bvarpi i}

${displaystyle avarpi +bvarpi i}$ for integers $a$ and $b$ . It has simple poles at Gaussian half-integer multiples of $ϖ$ , complex numbers of the form

{displaystyle {bigl (}a+{tfrac {1}{2}}{bigr )}varpi +{bigl (}b+{tfrac {1}{2}}{bigr )}varpi i}

${displaystyle {bigl (}a+{tfrac {1}{2}}{bigr )}varpi +{bigl (}b+{tfrac {1}{2}}{bigr )}varpi i}$ , with residues

{displaystyle (-1)^{a-b+1}i}

${displaystyle (-1)^{a-b+1}i}$ . The $cl$ function is reflected and offset from the $sl$ function,

{displaystyle operatorname {cl} z={operatorname {sl} }{bigl (}{tfrac {1}{2}}varpi -z{bigr )}}

${displaystyle operatorname {cl} z={operatorname {sl} }{bigl (}{tfrac {1}{2}}varpi -z{bigr )}}$ . It has zeros for arguments

{displaystyle {bigl (}a+{tfrac {1}{2}}{bigr )}varpi +bvarpi i}

${displaystyle {bigl (}a+{tfrac {1}{2}}{bigr )}varpi +bvarpi i}$ and poles for arguments

{displaystyle avarpi +{bigl (}b+{tfrac {1}{2}}{bigr )}varpi i,}

${displaystyle avarpi +{bigl (}b+{tfrac {1}{2}}{bigr )}varpi i,}$ with residues

{displaystyle (-1)^{a-b}i.}

${displaystyle (-1)^{a-b}i.}$

Also

{displaystyle operatorname {sl} z=operatorname {sl} wleftrightarrow z=(-1)^{m+n}w+(m+ni)varpi }

for some

{displaystyle m,nin mathbb {Z} }

${displaystyle m,nin mathbb {Z} }$ and

{displaystyle operatorname {sl} ((1pm i)z)=(1pm i){frac {operatorname {sl} z}{operatorname {sl} ‘z}}.}

The last formula is a special case of complex multiplication. Analogous formulas can be given for

{displaystyle operatorname {sl} ((n+mi)z)}

${displaystyle operatorname {sl} ((n+mi)z)}$ where

{displaystyle n+mi}

${displaystyle n+mi}$ is any Gaussian integer – the function

{displaystyle operatorname {sl} }

${displaystyle operatorname {sl} }$ has complex multiplication by

{displaystyle mathbb {Z} [i]}

$mathbb {Z} [i]$ .^[22]

There are also infinite series reflecting the distribution of the zeros and poles of $sl$ :^[23]^[24]

{displaystyle {frac {1}{operatorname {sl} z}}=sum _{(n,k)in mathbb {Z} ^{2}}{frac {(-1)^{n+k}}{z+nvarpi +kvarpi i}}}

{displaystyle operatorname {sl} z=-isum _{(n,k)in mathbb {Z} ^{2}}{frac {(-1)^{n+k}}{z+(n+1/2)varpi +(k+1/2)varpi i}}.}

The lemniscate sine as a ratio of entire functions[edit]

The

{displaystyle M}

The

{displaystyle N}

Since the lemniscate sine is a meromorphic function in the whole complex plane, it can be written as a ratio of entire functions. Gauss showed that $sl$ has the following product expansion, reflecting the distribution of its zeros and poles:^[25]

{displaystyle operatorname {sl} z={frac {M(z)}{N(z)}}}

where

{displaystyle M(z)=zprod _{alpha }left(1-{frac {z^{4}}{alpha ^{4}}}right),quad N(z)=prod _{beta }left(1-{frac {z^{4}}{beta ^{4}}}right).}

Here,

{displaystyle alpha }

$alpha$ and

{displaystyle beta }

$beta$ denote, respectively, the zeros and poles of $sl$ which are in the quadrant

{displaystyle operatorname {Re} z>0,operatorname {Im} zgeq 0}

${displaystyle operatorname {Re} z>0,operatorname {Im} zgeq 0}”/></span>. Gauss conjectured that <span class=$

{displaystyle ln N(varpi )=pi /2}

${displaystyle ln N(varpi )=pi /2}$ (this later turned out to be true) and commented that this “is most remarkable and a proof of this property promises the most serious increase in analysis”.^[26] Gauss expanded the products for

{displaystyle M}

$M$ and

{displaystyle N}

$N$ as infinite series. He also discovered several identities involving the functions

{displaystyle M}

$M$ and

{displaystyle N}

$N$ , such as

{displaystyle N(z)={frac {M((1+i)z)}{(1+i)M(z)}},}

{displaystyle N(2z)=M^{4}(z)+N^{4}(z)}

and

{displaystyle M(2z)=2M(z)N(z)N((1+i)z).}

Since the functions

{displaystyle M}

$M$ and

{displaystyle N}

$N$ are entire, their power series expansions converge everywhere in the complex plane:^[27]^[28]^[29]

{displaystyle M(z)=z-2{frac {z^{5}}{5!}}-36{frac {z^{9}}{9!}}+552{frac {z^{13}}{13!}}+cdots ,quad zin mathbb {C} }

{displaystyle N(z)=1+2{frac {z^{4}}{4!}}-4{frac {z^{8}}{8!}}+408{frac {z^{12}}{12!}}+cdots ,quad zin mathbb {C} .}

Pythagorean-like identity[edit]

Curves

x ² \oplus y ² = a

for various values of a. Negative

a

in green, positive

a

in blue,

a = \pm1

in red,

a = \infty

in black.

The lemniscate functions satisfy a Pythagorean-like identity:

{displaystyle operatorname {cl^{2}} z+operatorname {sl^{2}} z+operatorname {cl^{2}} z,operatorname {sl^{2}} z=1}

As a result, the parametric equation

{displaystyle (x,y)=(operatorname {cl} t,operatorname {sl} t)}

${displaystyle (x,y)=(operatorname {cl} t,operatorname {sl} t)}$ parametrizes the quartic curve

{displaystyle x^{2}+y^{2}+x^{2}y^{2}=1.}

${displaystyle x^{2}+y^{2}+x^{2}y^{2}=1.}$

This identity can alternately be rewritten:^[30]

{displaystyle {bigl (}1+operatorname {cl^{2}} z{bigr )}{bigl (}1+operatorname {sl^{2}} z{bigr )}=2}

{displaystyle operatorname {cl^{2}} z={frac {1-operatorname {sl^{2}} z}{1+operatorname {sl^{2}} z}},quad operatorname {sl^{2}} z={frac {1-operatorname {cl^{2}} z}{1+operatorname {cl^{2}} z}}}

Defining a tangent-sum operator as

{displaystyle aoplus bmathrel {:=} tan(arctan a+arctan b),}

${displaystyle aoplus bmathrel {:=} tan(arctan a+arctan b),}$ gives:

{displaystyle operatorname {cl^{2}} zoplus operatorname {sl^{2}} z=1.}

The functions

{displaystyle {tilde {operatorname {cl} }}}

${displaystyle {tilde {operatorname {cl} }}}$ and

{displaystyle {tilde {operatorname {sl} }}}

${displaystyle {tilde {operatorname {sl} }}}$ satisfy another Pythagorean-like identity:

{displaystyle left(int _{0}^{x}{tilde {operatorname {cl} }},t,mathrm {d} tright)^{2}+left(1-int _{0}^{x}{tilde {operatorname {sl} }},t,mathrm {d} tright)^{2}=1.}

Derivatives and integrals[edit]

The derivatives are as follows:

{displaystyle {begin{aligned}{frac {mathrm {d} }{mathrm {d} z}}operatorname {cl} z=operatorname {cl’} z&=-{bigl (}1+operatorname {cl^{2}} z{bigr )}operatorname {sl} z=-{frac {2operatorname {sl} z}{operatorname {sl} ^{2}z+1}}\operatorname {cl’^{2}} z&=1-operatorname {cl^{4}} z\[5mu]{frac {mathrm {d} }{mathrm {d} z}}operatorname {sl} z=operatorname {sl’} z&={bigl (}1+operatorname {sl^{2}} z{bigr )}operatorname {cl} z={frac {2operatorname {cl} z}{operatorname {cl} ^{2}z+1}}\operatorname {sl’^{2}} z&=1-operatorname {sl^{4}} zend{aligned}}}

The second derivatives of lemniscate sine and lemniscate cosine are their negative duplicated cubes:

{displaystyle {frac {mathrm {d} ^{2}}{mathrm {d} z^{2}}}operatorname {cl} z=-2operatorname {cl^{3}} z}

{displaystyle {frac {mathrm {d} ^{2}}{mathrm {d} z^{2}}}operatorname {sl} z=-2operatorname {sl^{3}} z}

The lemniscate functions can be integrated using the inverse tangent function:

{displaystyle int operatorname {cl} zmathop {mathrm {d} z} =arctan operatorname {sl} z+C}

{displaystyle int operatorname {sl} zmathop {mathrm {d} z} =-arctan operatorname {cl} z+C}

Argument sum and multiple identities[edit]

Like the trigonometric functions, the lemniscate functions satisfy argument sum and difference identities. The original identity used by Fagnano for bisection of the lemniscate was:^[31]

{displaystyle operatorname {sl} (u+v)={frac {operatorname {sl} u,operatorname {sl’} v+operatorname {sl} v,operatorname {sl’} u}{1+operatorname {sl^{2}} u,operatorname {sl^{2}} v}}}

The derivative and Pythagorean-like identities can be used to rework the identity used by Fagano in terms of $sl$ and $cl$ . Defining a tangent-sum operator

{displaystyle aoplus bmathrel {:=} tan(arctan a+arctan b)}

${displaystyle aoplus bmathrel {:=} tan(arctan a+arctan b)}$ and tangent-difference operator

{displaystyle aominus bmathrel {:=} aoplus (-b),}

${displaystyle aominus bmathrel {:=} aoplus (-b),}$ the argument sum and difference identities can be expressed as:^[32]

{displaystyle {begin{aligned}operatorname {cl} (u+v)&=operatorname {cl} u,operatorname {cl} vominus operatorname {sl} u,operatorname {sl} v={frac {operatorname {cl} u,operatorname {cl} v-operatorname {sl} u,operatorname {sl} v}{1+operatorname {sl} u,operatorname {cl} u,operatorname {sl} v,operatorname {cl} v}}\[2mu]operatorname {cl} (u-v)&=operatorname {cl} u,operatorname {cl} voplus operatorname {sl} u,operatorname {sl} v\[2mu]operatorname {sl} (u+v)&=operatorname {sl} u,operatorname {cl} voplus operatorname {cl} u,operatorname {sl} v={frac {operatorname {sl} u,operatorname {cl} v+operatorname {cl} u,operatorname {sl} v}{1-operatorname {sl} u,operatorname {cl} u,operatorname {sl} v,operatorname {cl} v}}\[2mu]operatorname {sl} (u-v)&=operatorname {sl} u,operatorname {cl} vominus operatorname {cl} u,operatorname {sl} vend{aligned}}}

These resemble their trigonometric analogs:

{displaystyle {begin{aligned}cos(upm v)&=cos u,cos vmp sin u,sin v\[6mu]sin(upm v)&=sin u,cos vpm cos u,sin vend{aligned}}}

In particular, to compute the complex-valued functions in real components,

{displaystyle {begin{aligned}operatorname {cl} (x+iy)&={frac {operatorname {cl} x-ioperatorname {sl} x,operatorname {sl} y,operatorname {cl} y}{operatorname {cl} y+ioperatorname {sl} x,operatorname {cl} x,operatorname {sl} y}}\[4mu]&={frac {operatorname {cl} x,operatorname {cl} yleft(1-operatorname {sl} ^{2}x,operatorname {sl} ^{2}yright)}{operatorname {cl} ^{2}y+operatorname {sl} ^{2}x,operatorname {cl} ^{2}x,operatorname {sl} ^{2}y}}-i{frac {operatorname {sl} x,operatorname {sl} yleft(operatorname {cl} ^{2}x+operatorname {cl} ^{2}yright)}{operatorname {cl} ^{2}y+operatorname {sl} ^{2}x,operatorname {cl} ^{2}x,operatorname {sl} ^{2}y}}\[12mu]operatorname {sl} (x+iy)&={frac {operatorname {sl} x+ioperatorname {cl} x,operatorname {sl} y,operatorname {cl} y}{operatorname {cl} y-ioperatorname {sl} x,operatorname {cl} x,operatorname {sl} y}}\[4mu]&={frac {operatorname {sl} x,operatorname {cl} yleft(1-operatorname {cl} ^{2}x,operatorname {sl} ^{2}yright)}{operatorname {cl} ^{2}y+operatorname {sl} ^{2}x,operatorname {cl} ^{2}x,operatorname {sl} ^{2}y}}+i{frac {operatorname {cl} x,operatorname {sl} yleft(operatorname {sl} ^{2}x+operatorname {cl} ^{2}yright)}{operatorname {cl} ^{2}y+operatorname {sl} ^{2}x,operatorname {cl} ^{2}x,operatorname {sl} ^{2}y}}end{aligned}}}

Bisection formulas:

{displaystyle operatorname {cl} ^{2}{tfrac {1}{2}}x={frac {1+operatorname {cl} x{sqrt {1+operatorname {sl} ^{2}x}}}{{sqrt {1+operatorname {sl} ^{2}x}}+1}}}

{displaystyle operatorname {sl} ^{2}{tfrac {1}{2}}x={frac {1-operatorname {cl} x{sqrt {1+operatorname {sl} ^{2}x}}}{{sqrt {1+operatorname {sl} ^{2}x}}+1}}}

Duplication formulas:^[33]

{displaystyle operatorname {cl} 2x={frac {-1+2,operatorname {cl} ^{2}x+operatorname {cl} ^{4}x}{1+2,operatorname {cl} ^{2}x-operatorname {cl} ^{4}x}}}

{displaystyle operatorname {sl} 2x=2,operatorname {sl} x,operatorname {cl} x{frac {1+operatorname {sl} ^{2}x}{1+operatorname {sl} ^{4}x}}}

Triplication formulas:^[33]

{displaystyle operatorname {cl} 3x={frac {-3,operatorname {cl} x+6,operatorname {cl} ^{5}x+operatorname {cl} ^{9}x}{1+6,operatorname {cl} ^{4}x-3,operatorname {cl} ^{8}x}}}

{displaystyle operatorname {sl} 3x={frac {color {red}{3},color {black}{operatorname {sl} x-,}color {green}{6},color {black}{operatorname {sl} ^{5}x-,}color {blue}{1},color {black}{operatorname {sl} ^{9}x}}{color {blue}{1},color {black}{+,},color {green}{6},color {black}{operatorname {sl} ^{4}x-,}color {red}{3},color {black}{operatorname {sl} ^{8}x}}}}

Note the “reverse symmetry” of the coefficients of numerator and denominator of

{displaystyle operatorname {sl} 3x}

${displaystyle operatorname {sl} 3x}$ . This phenomenon can be observed in multiplication formulas for

{displaystyle operatorname {sl} beta x}

${displaystyle operatorname {sl} beta x}$ where

{displaystyle beta =m+ni}

${displaystyle beta =m+ni}$ whenever

{displaystyle m,nin mathbb {Z} }

${displaystyle m,nin mathbb {Z} }$ and

{displaystyle m+n}

$m+n$ is odd.^[22]

Lemnatomic polynomials[edit]

Let

{displaystyle L}

$L$ be the lattice

{displaystyle L=mathbb {Z} (1+i)varpi +mathbb {Z} (1-i)varpi .}

Furthermore, let

{displaystyle K=mathbb {Q} (i)}

${displaystyle K=mathbb {Q} (i)}$ ,

{displaystyle {mathcal {O}}=mathbb {Z} [i]}

${displaystyle {mathcal {O}}=mathbb {Z} [i]}$ ,

{displaystyle zin mathbb {C} }

$zinmathbb{C}$ ,

{displaystyle beta =m+in}

${displaystyle beta =m+in}$ ,

{displaystyle gamma =m’+in’}

${displaystyle gamma =m'+in'}$ (where

{displaystyle m,n,m’,n’in mathbb {Z} }

${displaystyle m,n,m',n'in mathbb {Z} }$ ),

{displaystyle m+n}

$m+n$ be odd,

{displaystyle m’+n’}

${displaystyle m'+n'}$ be odd,

{displaystyle gamma equiv 1,operatorname {mod} ,2(1+i)}

${displaystyle gamma equiv 1,operatorname {mod} ,2(1+i)}$ and

{displaystyle operatorname {sl} beta z=M_{beta }(operatorname {sl} z)}

${displaystyle operatorname {sl} beta z=M_{beta }(operatorname {sl} z)}$ . Then

{displaystyle M_{beta }(x)=i^{varepsilon }x{frac {P_{beta }(x^{4})}{Q_{beta }(x^{4})}}}

for some coprime polynomials

{displaystyle P_{beta }(x),Q_{beta }(x)in {mathcal {O}}[x]}

${displaystyle P_{beta }(x),Q_{beta }(x)in {mathcal {O}}[x]}$
and some

{displaystyle varepsilon in {0,1,2,3}}

${displaystyle varepsilon in {0,1,2,3}}$ ^[34] where

{displaystyle xP_{beta }(x^{4})=prod _{gamma |beta }Lambda _{gamma }(x)}

and

{displaystyle Lambda _{beta }(x)=prod _{[alpha ]in ({mathcal {O}}/beta {mathcal {O}})^{times }}(x-operatorname {sl} alpha delta _{beta })}

where

{displaystyle delta _{beta }}

${displaystyle delta _{beta }}$ is any

{displaystyle beta }

$beta$ -torsion generator (i.e.

{displaystyle delta _{beta }in (1/beta )L}

${displaystyle delta _{beta }in (1/beta )L}$ and

{displaystyle [delta _{beta }]in (1/beta )L/L}

${displaystyle [delta _{beta }]in (1/beta )L/L}$ generates

{displaystyle (1/beta )L/L}

${displaystyle (1/beta )L/L}$ as an

{displaystyle {mathcal {O}}}

${mathcal {O}}$ -module). Examples of

{displaystyle beta }

$beta$ -torsion generators include

{displaystyle 2varpi /beta }

${displaystyle 2varpi /beta }$ and

{displaystyle (1+i)varpi /beta }

${displaystyle (1+i)varpi /beta }$ . The polynomial

{displaystyle Lambda _{beta }(x)in {mathcal {O}}[x]}

${displaystyle Lambda _{beta }(x)in {mathcal {O}}[x]}$ is called the

{displaystyle beta }

$beta$ -th lemnatomic polynomial. It is monic and is irreducible over

{displaystyle K}

$K$ . The lemnatomic polynomials are the “lemniscate analogs” of the cyclotomic polynomials,^[35]

{displaystyle Phi _{k}(x)=prod _{[a]in (mathbb {Z} /kmathbb {Z} )^{times }}(x-zeta _{k}^{a}).}

The

{displaystyle beta }

$beta$ -th lemnatomic polynomial

{displaystyle Lambda _{beta }(x)}

${displaystyle Lambda _{beta }(x)}$ is the minimal polynomial of

{displaystyle operatorname {sl} delta _{beta }}

${displaystyle operatorname {sl} delta _{beta }}$ in

{displaystyle K[x]}

$K[x]$ . For convenience, let

{displaystyle omega _{beta }=operatorname {sl} (2varpi /beta )}

${displaystyle omega _{beta }=operatorname {sl} (2varpi /beta )}$ and

{displaystyle {tilde {omega }}_{beta }=operatorname {sl} ((1+i)varpi /beta )}

${displaystyle {tilde {omega }}_{beta }=operatorname {sl} ((1+i)varpi /beta )}$ . So for example, the minimal polynomial of

{displaystyle omega _{5}}

${displaystyle omega _{5}}$ (and also of

{displaystyle {tilde {omega }}_{5}}

${displaystyle {tilde {omega }}_{5}}$ ) in

{displaystyle K[x]}

$K[x]$ is

{displaystyle Lambda _{5}(x)=x^{16}+52x^{12}-26x^{8}-12x^{4}+1,}

and^[36]

{displaystyle omega _{5}={sqrt[{4}]{-13+6{sqrt {5}}+2{sqrt {85-38{sqrt {5}}}}}}}

{displaystyle {tilde {omega }}_{5}={sqrt[{4}]{-13-6{sqrt {5}}+2{sqrt {85+38{sqrt {5}}}}}}}

(an equivalent expression is given in the table below). Another example is^[35]

{displaystyle Lambda _{-1+2i}(x)=x^{4}-1+2i}

which is the minimal polynomial of

{displaystyle omega _{-1+2i}}

${displaystyle omega _{-1+2i}}$ (and also of

{displaystyle {tilde {omega }}_{-1+2i}}

${displaystyle {tilde {omega }}_{-1+2i}}$ ) in

{displaystyle K[x].}

${displaystyle K[x].}$

{displaystyle p}

$p$ is prime and

{displaystyle beta }

$beta$ is positive and odd,^[38] then^[39]

{displaystyle operatorname {deg} Lambda _{beta }=beta ^{2}prod _{p|beta }left(1-{frac {1}{p}}right)left(1-{frac {(-1)^{(p-1)/2}}{p}}right)}

which can be compared to the cyclotomic analog

{displaystyle operatorname {deg} Phi _{k}=kprod _{p|k}left(1-{frac {1}{p}}right).}

Specific values[edit]

Just as for the trigonometric functions, values of the lemniscate functions can be computed for divisions of the lemniscate into $n$ parts of equal length, using only basic arithmetic and square roots, if and only if $n$ is of the form

{displaystyle n=2^{k}p_{1}p_{2}cdots p_{m}}

${displaystyle n=2^{k}p_{1}p_{2}cdots p_{m}}$ where $k$ is a non-negative integer and each $p i$ (if any) is a distinct Fermat prime.^[40] The expressions become unwieldy as $n$ grows. Below are the expressions for dividing the lemniscate

{displaystyle (x^{2}+y^{2})^{2}=x^{2}-y^{2}}

$(x^{2}+y^{2})^{2}=x^{2}-y^{2}$ into $n$ parts of equal length for some $n \leq 20$ .

Power series[edit]

The power series expansion of the lemniscate sine at the origin is^[41]

{displaystyle operatorname {sl} z=sum _{n=0}^{infty }a_{n}z^{n}=z-12{frac {z^{5}}{5!}}+3024{frac {z^{9}}{9!}}-4390848{frac {z^{13}}{13!}}+cdots ,quad |z|<{tfrac {varpi }{sqrt {2}}}}

where the coefficients

{displaystyle a_{n}}

$a_{n}$ are determined as follows:

{displaystyle nnot equiv 1{pmod {4}}implies a_{n}=0,}

{displaystyle a_{1}=1,,forall nin mathbb {N} _{0}:,a_{n+2}=-{frac {2}{(n+1)(n+2)}}sum _{i+j+k=n}a_{i}a_{j}a_{k}}

where

{displaystyle i+j+k=n}

${displaystyle i+j+k=n}$ stands for all three-term compositions of

{displaystyle n}

$n$ . For example, to evaluate

{displaystyle a_{13}}

$a_{{13}}$ , it can be seen that there are only six compositions of

{displaystyle 13-2=11}

${displaystyle 13-2=11}$ that give a nonzero contribution to the sum:

{displaystyle 11=9+1+1=1+9+1=1+1+9}

${displaystyle 11=9+1+1=1+9+1=1+1+9}$ and

{displaystyle 11=5+5+1=5+1+5=1+5+5}

${displaystyle 11=5+5+1=5+1+5=1+5+5}$ , so

{displaystyle a_{13}=-{tfrac {2}{12cdot 13}}(a_{9}a_{1}a_{1}+a_{1}a_{9}a_{1}+a_{1}a_{1}a_{9}+a_{5}a_{5}a_{1}+a_{5}a_{1}a_{5}+a_{1}a_{5}a_{5})=-{tfrac {11}{15600}}.}

The expansion can be equivalently written as^[42]

{displaystyle operatorname {sl} z=sum _{n=0}^{infty }p_{2n}{frac {z^{4n+1}}{(4n+1)!}},quad left|zright|<{frac {varpi }{sqrt {2}}}}

where

{displaystyle p_{n+2}=-12sum _{j=0}^{n}{binom {2n+2}{2j+2}}p_{n-j}sum _{k=0}^{j}{binom {2j+1}{2k+1}}p_{k}p_{j-k},quad p_{0}=1,,p_{1}=0.}

The power series expansion of

{displaystyle {tilde {operatorname {sl} }}}

${displaystyle {tilde {operatorname {sl} }}}$ at the origin is

{displaystyle {tilde {operatorname {sl} }},z=sum _{n=0}^{infty }alpha _{n}z^{n}=z-9{frac {z^{3}}{3!}}+153{frac {z^{5}}{5!}}-4977{frac {z^{7}}{7!}}+cdots ,quad left|zright|<{frac {varpi }{2}}}

where

{displaystyle alpha _{n}=0}

${displaystyle alpha _{n}=0}$ if

{displaystyle n}

$n$ is even and^[43]

{displaystyle alpha _{n}={sqrt {2}}{frac {pi }{varpi }}{frac {(-1)^{(n-1)/2}}{n!}}sum _{k=1}^{infty }{frac {(2kpi /varpi )^{n+1}}{cosh kpi }},quad left|alpha _{n}right|sim 2^{n+5/2}{frac {n+1}{varpi ^{n+2}}}}

{displaystyle n}

$n$ is odd.

The expansion can be equivalently written as^[44]

{displaystyle {tilde {operatorname {sl} }},z=sum _{n=0}^{infty }{frac {(-1)^{n}}{2^{n+1}}}left(sum _{l=0}^{n}2^{l}{binom {2n+2}{2l+1}}s_{l}t_{n-l}right){frac {z^{2n+1}}{(2n+1)!}},quad left|zright|<{frac {varpi }{2}}}

where

{displaystyle s_{n+2}=3s_{n+1}+24sum _{j=0}^{n}{binom {2n+2}{2j+2}}s_{n-j}sum _{k=0}^{j}{binom {2j+1}{2k+1}}s_{k}s_{j-k},quad s_{0}=1,,s_{1}=3,}

{displaystyle t_{n+2}=3t_{n+1}+3sum _{j=0}^{n}{binom {2n+2}{2j+2}}t_{n-j}sum _{k=0}^{j}{binom {2j+1}{2k+1}}t_{k}t_{j-k},quad t_{0}=1,,t_{1}=3.}

For the lemniscate cosine,^[45]

{displaystyle operatorname {cl} {z}=1-sum _{n=0}^{infty }(-1)^{n}left(sum _{l=0}^{n}2^{l}{binom {2n+2}{2l+1}}q_{l}r_{n-l}right){frac {z^{2n+2}}{(2n+2)!}}=1-2{frac {z^{2}}{2!}}+12{frac {z^{4}}{4!}}-216{frac {z^{6}}{6!}}+cdots ,quad left|zright|<{frac {varpi }{2}},}

{displaystyle {tilde {operatorname {cl} }},z=sum _{n=0}^{infty }(-1)^{n}2^{n}q_{n}{frac {z^{2n}}{(2n)!}}=1-3{frac {z^{2}}{2!}}+33{frac {z^{4}}{4!}}-819{frac {z^{6}}{6!}}+cdots ,quad left|zright|<{frac {varpi }{2}}}

where

{displaystyle r_{n+2}=3sum _{j=0}^{n}{binom {2n+2}{2j+2}}r_{n-j}sum _{k=0}^{j}{binom {2j+1}{2k+1}}r_{k}r_{j-k},quad r_{0}=1,,r_{1}=0,}

{displaystyle q_{n+2}={tfrac {3}{2}}q_{n+1}+6sum _{j=0}^{n}{binom {2n+2}{2j+2}}q_{n-j}sum _{k=0}^{j}{binom {2j+1}{2k+1}}q_{k}q_{j-k},quad q_{0}=1,,q_{1}={tfrac {3}{2}}.}

Relation to Weierstrass and Jacobi elliptic functions[edit]

The lemniscate functions are closely related to the Weierstrass elliptic function

{displaystyle wp (z;1,0)}

${displaystyle wp (z;1,0)}$ (the “lemniscatic case”), with invariants $g 2 = 1$ and $g 3 = 0$ . This lattice has fundamental periods

{displaystyle omega _{1}={sqrt {2}}varpi ,}

${displaystyle omega _{1}={sqrt {2}}varpi ,}$ and

{displaystyle omega _{2}=iomega _{1}}

${displaystyle omega _{2}=iomega _{1}}$ . The associated constants of the Weierstrass function are

{displaystyle e_{1}={tfrac {1}{2}}, e_{2}=0, e_{3}=-{tfrac {1}{2}}.}

${displaystyle e_{1}={tfrac {1}{2}}, e_{2}=0, e_{3}=-{tfrac {1}{2}}.}$

The related case of a Weierstrass elliptic function with $g 2 = a$ , $g 3 = 0$ may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: $a > 0$ and $a < 0$ . The period parallelogram is either a square or a rhombus. The Weierstrass elliptic function

{displaystyle wp (z;-1,0)}

${displaystyle wp (z;-1,0)}$ is called the “pseudolemniscatic case”.^[46]

The square of the lemniscate sine can be represented as

{displaystyle operatorname {sl} ^{2}z={frac {1}{wp (z;4,0)}}={frac {i}{2wp ((1-i)z;-1,0)}}={-2wp }{left({sqrt {2}}z+(i-1){frac {varpi }{sqrt {2}}};1,0right)}}

where the second and third argument of

{displaystyle wp }

$wp$ denote the lattice invariants $g 2$ and $g 3$ . Another representation is

{displaystyle operatorname {sl} ^{2}z={frac {varpi ^{2}}{wp (z/varpi ,i)}}}

where the second argument of

{displaystyle wp }

$wp$ denotes the period ratio

{displaystyle tau }

$tau$ .^[47] The lemniscate sine is a rational function in the Weierstrass elliptic function and its derivative:^[48]

{displaystyle operatorname {sl} z=(i-1){frac {wp ((1+i)z;1/4,0)}{wp ‘((1+i)z;1/4,0)}}}

where the second and third argument of

{displaystyle wp }

$wp$ denote the lattice invariants $g 2$ and $g 3$ . In terms of the period ratio

{displaystyle tau }

$tau$ , this becomes

{displaystyle operatorname {sl} z=2varpi (i-1){frac {wp ((1+i)z/(2varpi ),i)}{wp ‘((1+i)z/(2varpi ),i)}}.}

The lemniscate functions can also be written in terms of Jacobi elliptic functions. The Jacobi elliptic functions

{displaystyle operatorname {sn} }

${displaystyle operatorname {sn} }$ and

{displaystyle operatorname {cd} }

${displaystyle operatorname {cd} }$ with positive real elliptic modulus have an “upright” rectangular lattice aligned with real and imaginary axes. Alternately, the functions

{displaystyle operatorname {sn} }

${displaystyle operatorname {sn} }$ and

{displaystyle operatorname {cd} }

${displaystyle operatorname {cd} }$ with modulus $i$ (and

{displaystyle operatorname {sd} }

${displaystyle operatorname {sd} }$ and

{displaystyle operatorname {cn} }

${displaystyle operatorname {cn} }$ with modulus

{displaystyle 1/{sqrt {2}}}

$1/{sqrt {2}}$ ) have a square period lattice rotated 1/8 turn.^[49]^[50]

{displaystyle operatorname {sl} z=operatorname {sn} (z;i)=operatorname {sc} (z;{sqrt {2}})={{tfrac {1}{sqrt {2}}}operatorname {sd} }left({sqrt {2}}z;{tfrac {1}{sqrt {2}}}right)}

{displaystyle operatorname {cl} z=operatorname {cd} (z;i)=operatorname {dn} (z;{sqrt {2}})={operatorname {cn} }left({sqrt {2}}z;{tfrac {1}{sqrt {2}}}right)}

where the second arguments denote the elliptic modulus

{displaystyle k}

$k$ .

The functions

{displaystyle {tilde {operatorname {sl} }}}

${displaystyle {tilde {operatorname {sl} }}}$ and

{displaystyle {tilde {operatorname {cl} }}}

${displaystyle {tilde {operatorname {cl} }}}$ can also be expressed in terms of Jacobi elliptic functions:

{displaystyle {tilde {operatorname {sl} }},z=operatorname {cd} (z;i)operatorname {sd} (z;i)=operatorname {dn} (z;{sqrt {2}})operatorname {sn} (z;{sqrt {2}})={tfrac {1}{sqrt {2}}}operatorname {cn} left({sqrt {2}}z;{tfrac {1}{sqrt {2}}}right)operatorname {sn} left({sqrt {2}}z;{tfrac {1}{sqrt {2}}}right),}

{displaystyle {tilde {operatorname {cl} }},z=operatorname {cd} (z;i)operatorname {nd} (z;i)=operatorname {dn} (z;{sqrt {2}})operatorname {cn} (z;{sqrt {2}})=operatorname {cn} left({sqrt {2}}z;{tfrac {1}{sqrt {2}}}right)operatorname {dn} left({sqrt {2}}z;{tfrac {1}{sqrt {2}}}right).}

Relation to the modular lambda function[edit]

The lemniscate sine can be used for the computation of values of the modular lambda function:

{displaystyle prod _{k=1}^{n};{operatorname {sl} }{left({frac {2k-1}{2n+1}}{frac {varpi }{2}}right)}={sqrt[{8}]{frac {lambda ((2n+1)i)}{1-lambda ((2n+1)i)}}}}

For example:

{displaystyle {begin{aligned}&{operatorname {sl} }{bigl (}{tfrac {1}{14}}varpi {bigr )},{operatorname {sl} }{bigl (}{tfrac {3}{14}}varpi {bigr )},{operatorname {sl} }{bigl (}{tfrac {5}{14}}varpi {bigr )}\[7mu]&quad {}={sqrt[{8}]{frac {lambda (7i)}{1-lambda (7i)}}}={tan }{Bigl (}{{tfrac {1}{2}}operatorname {arccsc} }{Bigl (}{tfrac {1}{2}}{sqrt {8{sqrt {7}}+21}}+{tfrac {1}{2}}{sqrt {7}}+1{Bigr )}{Bigr )}\[18mu]&{operatorname {sl} }{bigl (}{tfrac {1}{18}}varpi {bigr )},{operatorname {sl} }{bigl (}{tfrac {3}{18}}varpi {bigr )},{operatorname {sl} }{bigl (}{tfrac {5}{18}}varpi {bigr )},{operatorname {sl} }{bigl (}{tfrac {7}{18}}varpi {bigr )}\[-3mu]&quad {}={sqrt[{8}]{frac {lambda (9i)}{1-lambda (9i)}}}={tan }{Biggl (}{frac {pi }{4}}-{arctan }{Biggl (}{frac {2{sqrt[{3}]{2{sqrt {3}}-2}}-2{sqrt[{3}]{2-{sqrt {3}}}}+{sqrt {3}}-1}{sqrt[{4}]{12}}}{Biggr )}{Biggr )}end{aligned}}}

Ramanujan’s cos/cosh identity[edit]

Ramanujan’s famous cos/cosh identity states that if

{displaystyle R(s)={frac {pi }{varpi {sqrt {2}}}}sum _{nin mathbb {Z} }{frac {cos(2npi s/varpi )}{cosh npi }},}

then^[43]

{displaystyle R(s)^{-2}+R(is)^{-2}=2,quad left|operatorname {Re} sright|<{frac {varpi }{2}},left|operatorname {Im} sright|<{frac {varpi }{2}}.}

There is a close relation between the lemniscate functions and

{displaystyle R(s)}

${displaystyle R(s)}$ . Indeed,^[43]^[51]

{displaystyle {tilde {operatorname {sl} }},s=-{frac {mathrm {d} }{mathrm {d} s}}R(s)quad left|operatorname {Im} sright|<{frac {varpi }{2}}}

{displaystyle {tilde {operatorname {cl} }},s={frac {mathrm {d} }{mathrm {d} s}}{sqrt {1-R(s)^{2}}},quad left|operatorname {Re} s-{frac {varpi }{2}}right|<{frac {varpi }{2}},,left|operatorname {Im} sright|<{frac {varpi }{2}}}

and

{displaystyle R(s)={frac {1}{sqrt {1+operatorname {sl} ^{2}s}}},quad left|operatorname {Im} sright|<{frac {varpi }{2}}.}

Continued fractions[edit]

For

{displaystyle zin mathbb {C} setminus {0}}

${displaystyle zin mathbb {C} setminus {0}}$ :^[52]

{displaystyle int _{0}^{infty }e^{-tz{sqrt {2}}}operatorname {cl} t,mathrm {d} t={cfrac {1/{sqrt {2}}}{z+{cfrac {a_{1}}{z+{cfrac {a_{2}}{z+{cfrac {a_{3}}{z+ddots }}}}}}}},quad a_{n}={frac {n^{2}}{4}}((-1)^{n+1}+3)}

{displaystyle int _{0}^{infty }e^{-tz{sqrt {2}}}operatorname {sl} toperatorname {cl} t,mathrm {d} t={cfrac {1/2}{z^{2}+b_{1}-{cfrac {a_{1}}{z^{2}+b_{2}-{cfrac {a_{2}}{z^{2}+b_{3}-ddots }}}}}},quad a_{n}=n^{2}(4n^{2}-1),,b_{n}=3(2n-1)^{2}}

Methods of computation[edit]

A fast algorithm, returning approximations to
${displaystyle operatorname {sl} x}$
${displaystyle operatorname {sl} x}$ (which get closer to
${displaystyle operatorname {sl} x}$
${displaystyle operatorname {sl} x}$ with increasing
${displaystyle N}$
$N$ ), is the following:^[53]

This is effectively using the arithmetic-geometric mean and is based on Landen’s transformations.^[54]

Several methods of computing

{displaystyle operatorname {sl} x}

${displaystyle operatorname {sl} x}$ involve first making the change of variables

{displaystyle pi x=varpi {tilde {x}}}

${displaystyle pi x=varpi {tilde {x}}}$ and then computing

{displaystyle operatorname {sl} (varpi {tilde {x}}/pi ).}

${displaystyle operatorname {sl} (varpi {tilde {x}}/pi ).}$

A hyperbolic series method:^[55]^[56]^[57]

{displaystyle operatorname {sl} left({frac {varpi }{pi }}xright)={frac {pi }{varpi }}sum _{nin mathbb {Z} }{frac {(-1)^{n}}{cosh(x-(n+1/2)pi )}},quad xin mathbb {C} }

{displaystyle {frac {1}{operatorname {sl} (varpi x/pi )}}={frac {pi }{varpi }}sum _{nin mathbb {Z} }{frac {(-1)^{n}}{{sinh }{left(x-npi right)}}}={frac {pi }{varpi }}sum _{nin mathbb {Z} }{frac {(-1)^{n}}{sin(x-npi i)}},quad xin mathbb {C} }

Fourier series method:^[58]

{displaystyle operatorname {sl} {Bigl (}{frac {varpi }{pi }}x{Bigr )}={frac {2pi }{varpi }}sum _{n=0}^{infty }{frac {(-1)^{n}sin((2n+1)x)}{cosh((n+1/2)pi )}},quad left|operatorname {Im} xright|<{frac {pi }{2}}}

{displaystyle operatorname {cl} left({frac {varpi }{pi }}xright)={frac {2pi }{varpi }}sum _{n=0}^{infty }{frac {cos((2n+1)x)}{cosh((n+1/2)pi )}},quad left|operatorname {Im} xright|<{frac {pi }{2}}}

{displaystyle {frac {1}{operatorname {sl} (varpi x/pi )}}={frac {pi }{varpi }}left({frac {1}{sin x}}-4sum _{n=0}^{infty }{frac {sin((2n+1)x)}{e^{(2n+1)pi }+1}}right),quad left|operatorname {Im} xright|

The lemniscate functions can be computed more rapidly by

{displaystyle {begin{aligned}operatorname {sl} {Bigl (}{frac {varpi }{pi }}x{Bigr )}&={frac {{theta _{1}}{left(x,e^{-pi }right)}}{{theta _{3}}{left(x,e^{-pi }right)}}},quad xin mathbb {C} \operatorname {cl} {Bigl (}{frac {varpi }{pi }}x{Bigr )}&={frac {{theta _{2}}{left(x,e^{-pi }right)}}{{theta _{4}}{left(x,e^{-pi }right)}}},quad xin mathbb {C} end{aligned}}}

where

{displaystyle {begin{aligned}theta _{1}(x,e^{-pi })&=sum _{nin mathbb {Z} }(-1)^{n+1}e^{-pi (n+1/2+x/pi )^{2}}=sum _{nin mathbb {Z} }(-1)^{n}e^{-pi (n+1/2)^{2}}sin((2n+1)x),\theta _{2}(x,e^{-pi })&=sum _{nin mathbb {Z} }(-1)^{n}e^{-pi (n+x/pi )^{2}}=sum _{nin mathbb {Z} }e^{-pi (n+1/2)^{2}}cos((2n+1)x),\theta _{3}(x,e^{-pi })&=sum _{nin mathbb {Z} }e^{-pi (n+x/pi )^{2}}=sum _{nin mathbb {Z} }e^{-pi n^{2}}cos 2nx,\theta _{4}(x,e^{-pi })&=sum _{nin mathbb {Z} }e^{-pi (n+1/2+x/pi )^{2}}=sum _{nin mathbb {Z} }(-1)^{n}e^{-pi n^{2}}cos 2nxend{aligned}}}

are the Jacobi theta functions.^[59]

Two other fast computation methods use the following sum and product series:

{displaystyle {text{sl}}{Bigl (}{frac {varpi }{pi }}x{Bigr )}=f{biggl (}{frac {4pi }{varpi }}sin xsum _{n=1}^{infty }{frac {cosh[(2n-1)pi ]}{cosh ^{2}[(2n-1)pi ]-cos ^{2}x}}{biggr )}}

{displaystyle {text{cl}}{Bigl (}{frac {varpi }{pi }}x{Bigr )}=f{biggl (}{frac {4pi }{varpi }}cos xsum _{n=1}^{infty }{frac {cosh[(2n-1)pi ]}{cosh ^{2}[(2n-1)pi ]-sin ^{2}x}}{biggr )}}

{displaystyle mathrm {sl} {Bigl (}{frac {varpi }{pi }}x{Bigr )}=2e^{-pi /4}sin xprod _{n=1}^{infty }{frac {1-2e^{-2npi }cos 2x+e^{-4npi }}{1+2e^{-(2n-1)pi }cos 2x+e^{-(4n-2)pi }}},quad xin mathbb {C} }

{displaystyle mathrm {cl} {Bigl (}{frac {varpi }{pi }}x{Bigr )}=2e^{-pi /4}cos xprod _{n=1}^{infty }{frac {1+2e^{-2npi }cos 2x+e^{-4npi }}{1-2e^{-(2n-1)pi }cos 2x+e^{-(4n-2)pi }}},quad xin mathbb {C} }

where

{displaystyle f(x)=tan(2arctan x)=2x/(1-x^{2}).}

${displaystyle f(x)=tan(2arctan x)=2x/(1-x^{2}).}$

Fourier series for the logarithm of the lemniscate sine:

{displaystyle ln operatorname {sl} left({frac {varpi }{pi }}xright)=ln 2-{frac {pi }{4}}+ln sin x+2sum _{n=1}^{infty }{frac {(-1)^{n}cos 2nx}{n(e^{npi }+(-1)^{n})}},quad left|operatorname {Im} xright|<{frac {pi }{2}}}

The following series identities were discovered by Ramanujan:^[60]

{displaystyle {frac {varpi ^{2}}{pi ^{2}operatorname {sl} ^{2}(varpi x/pi )}}={frac {1}{sin ^{2}x}}-{frac {1}{pi }}-8sum _{n=1}^{infty }{frac {ncos 2nx}{e^{2npi }-1}},quad left|operatorname {Im} xright|

{displaystyle arctan operatorname {sl} {Bigl (}{frac {varpi }{pi }}x{Bigr )}=2sum _{n=0}^{infty }{frac {sin((2n+1)x)}{(2n+1)cosh((n+1/2)pi )}},quad left|operatorname {Im} xright|<{frac {pi }{2}}}

The functions

{displaystyle {tilde {operatorname {sl} }}}

${displaystyle {tilde {operatorname {sl} }}}$ and

{displaystyle {tilde {operatorname {cl} }}}

${displaystyle {tilde {operatorname {cl} }}}$ analogous to

{displaystyle sin }

$sin$ and

{displaystyle cos }

$cos$ on the unit circle have the following Fourier and hyperbolic series expansions:^[43]^[51]^[61]

{displaystyle {tilde {operatorname {sl} }},s=2{sqrt {2}}{frac {pi ^{2}}{varpi ^{2}}}sum _{n=1}^{infty }{frac {nsin(2npi s/varpi )}{cosh npi }},quad left|operatorname {Im} sright|<{frac {varpi }{2}}}

{displaystyle {tilde {operatorname {cl} }},s={sqrt {2}}{frac {pi ^{2}}{varpi ^{2}}}sum _{n=0}^{infty }{frac {(2n+1)cos((2n+1)pi s/varpi )}{sinh((n+1/2)pi )}},quad left|operatorname {Im} sright|<{frac {varpi }{2}}}

{displaystyle {tilde {operatorname {sl} }},s={frac {pi ^{2}}{varpi ^{2}{sqrt {2}}}}sum _{nin mathbb {Z} }{frac {sinh(pi (n+s/varpi ))}{cosh ^{2}(pi (n+s/varpi ))}},quad sin mathbb {C} }

{displaystyle {tilde {operatorname {cl} }},s={frac {pi ^{2}}{varpi ^{2}{sqrt {2}}}}sum _{nin mathbb {Z} }{frac {(-1)^{n}}{cosh ^{2}(pi (n+s/varpi ))}},quad sin mathbb {C} }

Inverse functions[edit]

The inverse function of the lemniscate sine is the lemniscate arcsine, defined as

{displaystyle operatorname {arcsl} x=int _{0}^{x}{frac {mathrm {d} t}{sqrt {1-t^{4}}}}.}

It can also be represented by the hypergeometric function:

{displaystyle operatorname {arcsl} x=x,{}_{2}F_{1}left({tfrac {1}{2}},{tfrac {1}{4}};{tfrac {5}{4}};x^{4}right).}

The inverse function of the lemniscate cosine is the lemniscate arccosine. This function is defined by following expression:

{displaystyle operatorname {arccl} x=int _{x}^{1}{frac {mathrm {d} t}{sqrt {1-t^{4}}}}={tfrac {1}{2}}varpi -operatorname {arcsl} x}

For $x$ in the interval

{displaystyle -1leq xleq 1}

${displaystyle -1leq xleq 1}$ ,

{displaystyle operatorname {sl} operatorname {arcsl} x=x}

${displaystyle operatorname {sl} operatorname {arcsl} x=x}$ and

{displaystyle operatorname {cl} operatorname {arccl} x=x}

${displaystyle operatorname {cl} operatorname {arccl} x=x}$

For the halving of the lemniscate arc length these formulas are valid:

{displaystyle {begin{aligned}{operatorname {sl} }{bigl (}{tfrac {1}{2}}operatorname {arcsl} x{bigr )}&={sin }{bigl (}{tfrac {1}{2}}arcsin x{bigr )},{operatorname {sech} }{bigl (}{tfrac {1}{2}}operatorname {arsinh} x{bigr )}\{operatorname {sl} }{bigl (}{tfrac {1}{2}}operatorname {arcsl} x{bigr )}^{2}&={tan }{bigl (}{tfrac {1}{4}}arcsin x^{2}{bigr )}end{aligned}}}

Expression using elliptic integrals[edit]

The lemniscate arcsine and the lemniscate arccosine can also be expressed by the Legendre-Form:

These functions can be displayed directly by using the incomplete elliptic integral of the first kind:

{displaystyle operatorname {arcsl} x={frac {1}{sqrt {2}}}Fleft({arcsin }{frac {{sqrt {2}}x}{sqrt {1+x^{2}}}};{frac {1}{sqrt {2}}}right)}

{displaystyle operatorname {arcsl} x=2({sqrt {2}}-1)Fleft({arcsin }{frac {({sqrt {2}}+1)x}{{sqrt {1+x^{2}}}+1}};({sqrt {2}}-1)^{2}right)}

The arc lengths of the lemniscate can also be expressed by only using the arc lengths of ellipses (calculated by elliptic integrals of the second kind):

{displaystyle {begin{aligned}operatorname {arcsl} x={}&{frac {2+{sqrt {2}}}{2}}Eleft({arcsin }{frac {({sqrt {2}}+1)x}{{sqrt {1+x^{2}}}+1}};({sqrt {2}}-1)^{2}right)\[5mu]& -Eleft({arcsin }{frac {{sqrt {2}}x}{sqrt {1+x^{2}}}};{frac {1}{sqrt {2}}}right)+{frac {x{sqrt {1-x^{2}}}}{{sqrt {2}}(1+x^{2}+{sqrt {1+x^{2}}})}}end{aligned}}}

The lemniscate arccosine has this expression:

{displaystyle operatorname {arccl} x={frac {1}{sqrt {2}}}Fleft(arccos x;{frac {1}{sqrt {2}}}right)}

Use in integration[edit]

The lemniscate arcsine can be used to integrate many functions. Here is a list of important integrals (the constants of integration are omitted):

{displaystyle int {frac {1}{sqrt {1-x^{4}}}},mathrm {d} x=operatorname {arcsl} x}

{displaystyle int {frac {1}{sqrt {(x^{2}+1)(2x^{2}+1)}}},mathrm {d} x={operatorname {arcsl} }{frac {x}{sqrt {x^{2}+1}}}}

{displaystyle int {frac {1}{sqrt {x^{4}+6x^{2}+1}}},mathrm {d} x={operatorname {arcsl} }{frac {{sqrt {2}}x}{sqrt {{sqrt {x^{4}+6x^{2}+1}}+x^{2}+1}}}}

{displaystyle int {frac {1}{sqrt {x^{4}+1}}},mathrm {d} x={{sqrt {2}}operatorname {arcsl} }{frac {x}{sqrt {{sqrt {x^{4}+1}}+1}}}}

{displaystyle int {frac {1}{sqrt[{4}]{(1-x^{4})^{3}}}},mathrm {d} x={{sqrt {2}}operatorname {arcsl} }{frac {x}{sqrt {1+{sqrt {1-x^{4}}}}}}}

{displaystyle int {frac {1}{sqrt[{4}]{(x^{4}+1)^{3}}}},mathrm {d} x={operatorname {arcsl} }{frac {x}{sqrt[{4}]{x^{4}+1}}}}

{displaystyle int {frac {1}{sqrt[{4}]{(1-x^{2})^{3}}}},mathrm {d} x={2operatorname {arcsl} }{frac {x}{1+{sqrt {1-x^{2}}}}}}

{displaystyle int {frac {1}{sqrt[{4}]{(x^{2}+1)^{3}}}},mathrm {d} x={2operatorname {arcsl} }{frac {x}{{sqrt {x^{2}+1}}+1}}}

{displaystyle int {frac {1}{sqrt[{4}]{(ax^{2}+bx+c)^{3}}}},mathrm {d} x={{frac {2{sqrt {2}}}{sqrt[{4}]{4a^{2}c-ab^{2}}}}operatorname {arcsl} }{frac {2ax+b}{{sqrt {4a(ax^{2}+bx+c)}}+{sqrt {4ac-b^{2}}}}}}

{displaystyle int {sqrt {operatorname {sech} x}},mathrm {d} x={2operatorname {arcsl} }tanh {tfrac {1}{2}}x}

{displaystyle int {sqrt {sec x}},mathrm {d} x={2operatorname {arcsl} }tan {tfrac {1}{2}}x}

Hyperbolic lemniscate functions[edit]

The hyperbolic lemniscate sine (red) and hyperbolic lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric tangent (pale dashed red).

The hyperbolic lemniscate sine in the complex plane. Dark areas represent zeros and bright areas represent poles. The complex argument is represented by varying hue.

For convenience, let

{displaystyle sigma ={sqrt {2}}varpi }

${displaystyle sigma ={sqrt {2}}varpi }$ .

{displaystyle sigma }

$sigma$ is the “squircular” analog of

{displaystyle pi }

$pi$ (see below). The decimal expansion of

{displaystyle sigma }

$sigma$ (i.e.

{displaystyle 3.7081ldots }

${displaystyle 3.7081ldots }$ ^[62]) appears in entry 34e of chapter 11 of Ramanujan’s second notebook.^[63]

The hyperbolic lemniscate sine ( $slh$ ) and cosine ( $clh$ ) can be defined as inverses of elliptic integrals as follows:

{displaystyle zmathrel {overset {*}{=}} int _{0}^{operatorname {slh} z}{frac {mathrm {d} t}{sqrt {1+t^{4}}}}=int _{operatorname {clh} z}^{infty }{frac {mathrm {d} t}{sqrt {1+t^{4}}}}}

where in

{displaystyle (*)}

$(*)$ ,

{displaystyle z}

$z$ is in the square with corners

{displaystyle {sigma /2,sigma i/2,-sigma /2,-sigma i/2}}

${displaystyle {sigma /2,sigma i/2,-sigma /2,-sigma i/2}}$ . Beyond that square, the functions can be analytically continued to meromorphic functions in the whole complex plane.

The complete integral has the value:

{displaystyle int _{0}^{infty }{frac {mathrm {d} t}{sqrt {t^{4}+1}}}={tfrac {1}{4}}mathrm {B} {bigl (}{tfrac {1}{4}},{tfrac {1}{4}}{bigr )}={frac {sigma }{2}}=1.85407;46773;01371ldots }

Therefore, the two defined functions have following relation to each other:

{displaystyle operatorname {slh} z={operatorname {clh} }{{Bigl (}{frac {sigma }{2}}-z{Bigr )}}}

The product of hyperbolic lemniscate sine and hyperbolic lemniscate cosine is equal to one:

{displaystyle operatorname {slh} z,operatorname {clh} z=1}

The functions

{displaystyle operatorname {slh} }

${displaystyle operatorname {slh} }$ and

{displaystyle operatorname {clh} }

${displaystyle operatorname {clh} }$ have a square period lattice with fundamental periods

{displaystyle {sigma ,sigma i}}

${displaystyle {sigma ,sigma i}}$ .

The hyperbolic lemniscate functions can be expressed in terms of lemniscate sine and lemniscate cosine:

{displaystyle operatorname {slh} {bigl (}{sqrt {2}}z{bigr )}={frac {(1+operatorname {cl} ^{2}z)operatorname {sl} z}{{sqrt {2}}operatorname {cl} z}}}

{displaystyle operatorname {clh} {bigl (}{sqrt {2}}z{bigr )}={frac {(1+operatorname {sl} ^{2}z)operatorname {cl} z}{{sqrt {2}}operatorname {sl} z}}}

But there is also a relation to the Jacobi elliptic functions with the elliptic modulus one by square root of two:

{displaystyle operatorname {slh} z={frac {operatorname {sn} (z;1/{sqrt {2}})}{operatorname {cd} (z;1/{sqrt {2}})}}}

{displaystyle operatorname {clh} z={frac {operatorname {cd} (z;1/{sqrt {2}})}{operatorname {sn} (z;1/{sqrt {2}})}}}

The hyperbolic lemniscate sine has following imaginary relation to the lemniscate sine:

{displaystyle operatorname {slh} z={frac {1-i}{sqrt {2}}}operatorname {sl} left({frac {1+i}{sqrt {2}}}zright)={frac {operatorname {sl} left({sqrt[{4}]{-1}}zright)}{sqrt[{4}]{-1}}}}

This is analogous to the relationship between hyperbolic and trigonometric sine:

{displaystyle sinh z=-isin(iz)={frac {sin left({sqrt[{2}]{-1}}zright)}{sqrt[{2}]{-1}}}}

In a quartic Fermat curve

{displaystyle x^{4}+y^{4}=1}

${displaystyle x^{4}+y^{4}=1}$ (sometimes called a squircle) the hyperbolic lemniscate sine and cosine are analogous to the tangent and cotangent functions in a unit circle

{displaystyle x^{2}+y^{2}=1}

$x^2 + y^2 = 1$ (the quadratic Fermat curve). If the origin and a point on the curve are connected to each other by a line $L$ , the hyperbolic lemniscate sine of twice the enclosed area between this line and the x-axis is the y-coordinate of the intersection of $L$ with the line

{displaystyle x=1}

$x=1$ .^[65] Just as

{displaystyle pi }

$pi$ is the area enclosed by the circle

{displaystyle x^{2}+y^{2}=1}

$x^{2}+y^{2}=1$ , the area enclosed by the squircle

{displaystyle x^{4}+y^{4}=1}

${displaystyle x^{4}+y^{4}=1}$ is

{displaystyle sigma }

$sigma$ . Moreover,

{displaystyle M(1,1/{sqrt {2}})={frac {pi }{sigma }}}

where

{displaystyle M}

$M$ is the arithmetic–geometric mean.

The hyperbolic lemniscate sine satisfies the argument addition identity:

{displaystyle operatorname {slh} (a+b)={frac {operatorname {slh} aoperatorname {slh} ‘b+operatorname {slh} boperatorname {slh} ‘a}{1-operatorname {slh} ^{2}a,operatorname {slh} ^{2}b}}}

When

{displaystyle x}

$x$ is real, the derivative can be expressed in this way:

{displaystyle {frac {mathrm {d} }{mathrm {d} x}}operatorname {slh} x={sqrt {1+operatorname {slh} ^{4}x}}.}

Number theory[edit]

In algebraic number theory, every finite abelian extension of the Gaussian rationals

{displaystyle mathbb {Q} (i)}

${mathbb {Q}}(i)$ is a subfield of

{displaystyle mathbb {Q} (i,omega _{n})}

${displaystyle mathbb {Q} (i,omega _{n})}$ for some positive integer

{displaystyle n}

$n$ .^[35]^[66] This is analogous to the Kronecker–Weber theorem for the rational numbers

{displaystyle mathbb {Q} }

$mathbb {Q}$ which is based on division of the circle – in particular, every finite abelian extension of

{displaystyle mathbb {Q} }

$mathbb {Q}$ is a subfield of

{displaystyle mathbb {Q} (zeta _{n})}

${displaystyle mathbb {Q} (zeta _{n})}$ for some positive integer

{displaystyle n}

$n$ . Both are special cases of Kronecker’s Jugendtraum, which became Hilbert’s twelfth problem.

The field

{displaystyle mathbb {Q} (i,operatorname {sl} (varpi /n))}

${displaystyle mathbb {Q} (i,operatorname {sl} (varpi /n))}$ (for positive odd

{displaystyle n}

$n$ ) is the extension of

{displaystyle mathbb {Q} (i)}

${mathbb {Q}}(i)$ generated by the

{displaystyle x}

$x$ – and

{displaystyle y}

$y$ -coordinates of the

{displaystyle (1+i)n}

${displaystyle (1+i)n}$ -torsion points on the elliptic curve

{displaystyle y^{2}=4x^{3}+x}

${displaystyle y^{2}=4x^{3}+x}$ .^[66]

Hurwitz numbers[edit]

The Bernoulli numbers

{displaystyle mathrm {B} _{n}}

${displaystyle mathrm {B} _{n}}$ can be defined by

{displaystyle mathrm {B} _{n}=lim _{zto 0}{frac {mathrm {d} ^{n}}{mathrm {d} z^{n}}}{frac {z}{e^{z}-1}},quad ngeq 0}

and appear in

{displaystyle sum _{kin mathbb {Z} setminus {0}}{frac {1}{k^{2n}}}=(-1)^{n-1}mathrm {B} _{2n}{frac {(2pi )^{2n}}{(2n)!}}=2zeta (2n),quad ngeq 1}

where

{displaystyle zeta }

$zeta$ is the Riemann zeta function.

The Hurwitz numbers

{displaystyle mathrm {H} _{n},}

${displaystyle mathrm {H} _{n},}$ named after Adolf Hurwitz, are the “lemniscate analogs” of the Bernoulli numbers. They can be defined by^[67]^[68]

{displaystyle mathrm {H} _{n}=-lim _{zto 0}{frac {mathrm {d} ^{n}}{mathrm {d} z^{n}}}zzeta (z;1/4,0),quad ngeq 4}

where

{displaystyle zeta (cdot ;1/4,0)}

${displaystyle zeta (cdot ;1/4,0)}$ is the Weierstrass zeta function with lattice invariants

{displaystyle 1/4}

$1/4$ and

{displaystyle 0}

${displaystyle 0}$ . They appear in

{displaystyle sum _{zin mathbb {Z} [i]setminus {0}}{frac {1}{z^{4n}}}=mathrm {H} _{4n}{frac {(2varpi )^{4n}}{(4n)!}}=G_{4n}(i),quad ngeq 1}

where

{displaystyle mathbb {Z} [i]}

$mathbb {Z} [i]$ are the Gaussian integers and

{displaystyle G_{4n}}

${displaystyle G_{4n}}$ are the Eisenstein series of weight

{displaystyle 4n}

$4n$ , and in

{displaystyle displaystyle {begin{array}{ll}displaystyle sum _{n=1}^{infty }{dfrac {n^{k}}{e^{2pi n}-1}}={begin{cases}{dfrac {1}{24}}-{dfrac {1}{8pi }}&{text{if}} k=1\{dfrac {mathrm {B} _{k+1}}{2k+2}}&{text{if}} kequiv 1,(mathrm {mod} ,4) {text{and}} kgeq 5\{dfrac {mathrm {B} _{k+1}}{2k+2}}+{dfrac {mathrm {H} _{k+1}}{2k+2}}left({dfrac {varpi }{pi }}right)^{k+1}&{text{if}} kequiv 3,(mathrm {mod} ,4) {text{and}} kgeq 3.\end{cases}}end{array}}}

The Hurwitz numbers can also be determined as follows:

{displaystyle mathrm {H} _{4}=1/10}

${displaystyle mathrm {H} _{4}=1/10}$ ,

{displaystyle mathrm {H} _{4n}={frac {3}{(2n-3)(16n^{2}-1)}}sum _{k=1}^{n-1}{binom {4n}{4k}}(4k-1)(4(n-k)-1)mathrm {H} _{4k}mathrm {H} _{4(n-k)},quad ngeq 2}

and

{displaystyle mathrm {H} _{n}=0}

${displaystyle mathrm {H} _{n}=0}$ if

{displaystyle n}

$n$ is not a multiple of

{displaystyle 4}

$4$ .^[69] This yields^[67]

{displaystyle mathrm {H} _{8}={frac {3}{10}},,mathrm {H} _{12}={frac {567}{130}},,mathrm {H} _{16}={frac {43,659}{170}},,ldots }

Also^[70]

{displaystyle operatorname {denom} mathrm {H} _{4n}=2prod _{(p-1)|4n}p}

where

{displaystyle pin mathbb {P} }

${displaystyle pin mathbb {P} }$ such that

{displaystyle pequiv 1,({text{mod}},4),}

${displaystyle pequiv 1,({text{mod}},4),}$
just as

{displaystyle operatorname {denom} mathrm {B} _{2n}=prod _{(p-1)|2n}p}

where

{displaystyle pin mathbb {P} }

${displaystyle pin mathbb {P} }$ (by the von Staudt–Clausen theorem).

In fact, the von Staudt–Clausen theorem states that

{displaystyle mathrm {B} _{2n}+sum _{(p-1)|2n}{frac {1}{p}}in mathbb {Z} ,quad ngeq 1}

(sequence A000146 in the OEIS) where

{displaystyle p}

$p$ is any prime, and an analogous theorem holds for the Hurwitz numbers: suppose that

{displaystyle ain mathbb {Z} }

$ain {mathbb {Z}}$ is odd,

{displaystyle bin mathbb {Z} }

$bin {mathbb {Z}}$ is even,

{displaystyle p}

$p$ is a prime such that

{displaystyle pequiv 1,(mathrm {mod} ,4)}

${displaystyle pequiv 1,(mathrm {mod} ,4)}$ ,

{displaystyle p=a^{2}+b^{2}}

${displaystyle p=a^{2}+b^{2}}$ (see Fermat’s theorem on sums of two squares) and

{displaystyle aequiv b+1,(mathrm {mod} ,4)}

${displaystyle aequiv b+1,(mathrm {mod} ,4)}$ . Then for any given

{displaystyle p}

$p$ ,

{displaystyle a=a_{p}}

${displaystyle a=a_{p}}$ is uniquely determined and^[67]

{displaystyle mathrm {H} _{4n}-{frac {1}{2}}-sum _{(p-1)|4n}{frac {(2a_{p})^{4n/(p-1)}}{p}}mathrel {overset {text{def}}{=}} mathrm {G} _{n}in mathbb {Z} ,quad ngeq 1,}

{displaystyle operatorname {sl} z=sum _{n=0}^{infty }k_{4n+1}{frac {z^{4n+1}}{(4n+1)!}},quad left|zright|<{frac {varpi }{sqrt {2}}}implies k_{p}equiv 2a_{p},({text{mod}},p).}

The sequence of the integers

{displaystyle mathrm {G} _{n}}

${displaystyle mathrm {G} _{n}}$ starts with

{displaystyle 0,-1,5,253,ldots .}

${displaystyle 0,-1,5,253,ldots .}$ ^[67]

Let

{displaystyle ngeq 2}

$ngeq 2$ . If

{displaystyle 4n+1}

$4n+1$ is a prime, then

{displaystyle mathrm {G} _{n}equiv 1,(mathrm {mod} ,4)}

${displaystyle mathrm {G} _{n}equiv 1,(mathrm {mod} ,4)}$ . If

{displaystyle 4n+1}

$4n+1$ is not a prime, then

{displaystyle mathrm {G} _{n}equiv 3,(mathrm {mod} ,4)}

${displaystyle mathrm {G} _{n}equiv 3,(mathrm {mod} ,4)}$ .^[71]

Some authors instead define the Hurwitz numbers as

{displaystyle mathrm {H} _{n}’=mathrm {H} _{4n}}

${displaystyle mathrm {H} _{n}'=mathrm {H} _{4n}}$ .

Appearances in Laurent series[edit]

The Hurwitz numbers appear in several Laurent series expansions related to the lemniscate functions:^[72]

{displaystyle {begin{aligned}operatorname {sl} ^{2}z&=sum _{n=1}^{infty }{frac {2^{4n}(1-(-1)^{n}2^{2n})mathrm {H} _{4n}}{4n}}{frac {z^{4n-2}}{(4n-2)!}},quad left|zright|<{frac {varpi }{sqrt {2}}}\{frac {operatorname {sl} 'z}{operatorname {sl} {z}}}&={frac {1}{z}}-sum _{n=1}^{infty }{frac {2^{4n}(2-(-1)^{n}2^{2n})mathrm {H} _{4n}}{4n}}{frac {z^{4n-1}}{(4n-1)!}},quad left|zright|<{frac {varpi }{sqrt {2}}}\{frac {1}{operatorname {sl} z}}&={frac {1}{z}}-sum _{n=1}^{infty }{frac {2^{2n}((-1)^{n}2-2^{2n})mathrm {H} _{4n}}{4n}}{frac {z^{4n-1}}{(4n-1)!}},quad left|zright|

Analogously, in terms of the Bernoulli numbers:

{displaystyle {frac {1}{sinh ^{2}z}}={frac {1}{z^{2}}}-sum _{n=1}^{infty }{frac {2^{2n}mathrm {B} _{2n}}{2n}}{frac {z^{2n-2}}{(2n-2)!}},quad left|zright|

World map projections[edit]

The Peirce quincuncial projection, designed by Charles Sanders Peirce of the US Coast Survey in the 1870s, is a world map projection based on the inverse lemniscate sine of stereographically projected points (treated as complex numbers).^[73]

When lines of constant real or imaginary part are projected onto the complex plane via the hyperbolic lemniscate sine, and thence stereographically projected onto the sphere (see Riemann sphere), the resulting curves are spherical conics, the spherical analog of planar ellipses and hyperbolas.^[74] Thus the lemniscate functions (and more generally, the Jacobi elliptic functions) provide a parametrization for spherical conics.

A conformal map projection from the globe onto the 6 square faces of a cube can also be defined using the lemniscate functions.^[75] Because many partial differential equations can be effectively solved by conformal mapping, this map from sphere to cube is convenient for atmospheric modeling.^[76]

External links[edit]

References[edit]

Abel, Niels Henrik (1827–1828) “Recherches sur les fonctions elliptiques” [Research on elliptic functions] (in French). Crelle’s Journal.
Part 1. 1827. 2 (2): 101–181. doi:10.1515/crll.1827.2.101.
Part 2. 1828. 3 (3): 160–190. doi:10.1515/crll.1828.3.160.
Adams, Oscar Sherman (1925). Elliptic functions applied to conformal world maps (PDF). US Government Printing Office.
Ayoub, Raymond (1984). “The Lemniscate and Fagnano’s Contributions to Elliptic Integrals”. Archive for History of Exact Sciences. 29 (2): 131–149. doi:10.1007/BF00348244.
Berndt, Bruce C. (1994). Ramanujan’s Notebooks Part IV (First ed.). Springer. ISBN 978-1-4612-6932-8.
Bottazzini, Umberto; Gray, Jeremy (2013). Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory. Springer. doi:10.1007/978-1-4614-5725-1.
Carlson, Billie C. (2010). “19. Elliptic Integrals”. In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Cox, David Archibald (January 1984). “The Arithmetic-Geometric Mean of Gauss”. L’Enseignement Mathématique. 30 (2): 275–330.
Cox, David Archibald; Shurman, Jerry (2005). “Geometry and number theory on clovers” (PDF). The American Mathematical Monthly. 112 (8): 682–704. doi:10.1080/00029890.2005.11920241.
Cox, David Archibald (2012). “The Lemniscate”. Galois Theory. Wiley. pp. 463–514. doi:10.1002/9781118218457.ch15.
Cox, David Archibald (2013). Primes of the Form $x 2 + ny 2$ (Second ed.). Wiley.
Cox, David Archibald; Hyde, Trevor (2014). “The Galois theory of the lemniscate” (PDF). Journal of Number Theory. 135: 43–59. arXiv:1208.2653. doi:10.1016/j.jnt.2013.08.006.
Enneper, Alfred (1890) [1st ed. 1876]. “Note III: Historische Notizen über geometrische Anwendungen elliptischer Integrale.” [Historical notes on geometric applications of elliptic integrals]. Elliptische Functionen, Theorie und Geschichte (in German). Nebert. pp. 524–547.
Euler, Leonhard (1761). “Observationes de comparatione arcuum curvarum irrectificibilium” [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E252. (Figures)
Euler, Leonhard (1786). “De miris proprietatibus curvae elasticae sub aequatione ${textstyle y=int xxmathop {mathrm {d} x} {big /}{sqrt {1-x^{4}}}}$
${textstyle y=int xxmathop {mathrm {d} x} {big /}{sqrt {1-x^{4}}}}$ contentae” [On the amazing properties of elastic curves contained in equation
${textstyle y=int xxmathop {mathrm {d} x} {big /}{sqrt {1-x^{4}}}}$
${textstyle y=int xxmathop {mathrm {d} x} {big /}{sqrt {1-x^{4}}}}$ ]. Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin). 1782 (2): 34–61. E 605.
Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8.
Fagnano, Giulio Carlo (1718–1723) “Metodo per misurare la lemniscata” [Method for measuring the lemniscate]. Giornale de’ letterati d’Italia (in Italian).
“Schediasma primo” [Part 1]. 1718. 29: 258–269.
“Giunte al primo schediasma” [Addendum to part 1]. 1723. 34: 197–207.
“Schediasma secondo” [Part 2]. 1718. 30: 87–111.
Reprinted as Fagnano (1850). “32–34. Metodo per misurare la lemniscata”. Opere Matematiche, vol. 2. Allerighi e Segati. pp. 293–313. (Figures)
Gauss, Carl Friedrich (1917). Werke (Band X, Abteilung I) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
Gómez-Molleda, M. A.; Lario, Joan-C. (2019). “Ruler and Compass Constructions of the Equilateral Triangle and Pentagon in the Lemniscate Curve”. The Mathematical Intelligencer. 41 (4): 17–21. doi:10.1007/s00283-019-09892-w.
Greenhill, Alfred George (1892). The Applications of Elliptic Functions. MacMillan.
Guyou, Émile (1887). “Nouveau système de projection de la sphère: Généralisation de la projection de Mercator” [New system of projection of the sphere]. Annales Hydrographiques. Série 2 (in French). 9: 16–35.
Houzel, Christian (1978). “Fonctions elliptiques et intégrales abéliennes” [Elliptic functions and Abelian integrals]. In Dieudonné, Jean (ed.). Abrégé d’histoire des mathématiques, 1700–1900. II (in French). Hermann. pp. 1–113.
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.
Kubota, Tomio (1964). “Some arithmetical applications of an elliptic function”. Crelle’s Journal. 214/215: 141–145. doi:10.1515/crll.1964.214-215.141.
Langer, Joel C.; Singer, David A. (2010). “Reflections on the Lemniscate of Bernoulli: The Forty-Eight Faces of a Mathematical Gem” (PDF). Milan Journal of Mathematics. 78 (2): 643–682. doi:10.1007/s00032-010-0124-5.
Langer, Joel C.; Singer, David A. (2011). “The lemniscatic chessboard”. Forum Geometricorum. 11: 183–199.
Lawden, Derek Frank (1989). Elliptic Functions and Applications. Applied Mathematical Sciences. Vol. 80. Springer-Verlag. doi:10.1007/978-1-4757-3980-0.
Lee, Laurence Patrick (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monograph. Vol. 16. University of Toronto Press.
Lemmermeyer, Franz (2000). Reciprocity Laws: From Euler to Eisenstein. Springer. ISBN 3-540-66957-4.
Levien, Raph (2008). The elastica: a mathematical history (PDF) (Technical report). University of California at Berkeley. UCB/EECS-2008-103.
Levin, Aaron (2006). “A Geometric Interpretation of an Infinite Product for the Lemniscate Constant”. The American Mathematical Monthly. 113 (6): 510–520. doi:10.2307/27641976.
Lindqvist, Peter; Peetre, Jaak (2001). “Two Remarkable Identities, Called Twos, for Inverses to Some Abelian Integrals” (PDF). The American Mathematical Monthly. 108 (5): 403–410. doi:10.1080/00029890.2001.11919766.
Markushevich, Aleksei Ivanovich (1966). The Remarkable Sine Functions. Elsevier.
Markushevich, Aleksei Ivanovich (1992). Introduction to the Classical Theory of Abelian Functions. Translations of Mathematical Monographs. Vol. 96. American Mathematical Society. doi:10.1090/mmono/096.
McGregor, John L. (2005). C-CAM: Geometric Aspects and Dynamical Formulation (Technical report). CSIRO Atmospheric Research. 70.
McKean, Henry; Moll, Victor (1999). Elliptic Curves: Function Theory, Geometry, Arithmetic. Cambridge. ISBN 9780521582285.
Milne-Thomson, Louis Melville (1964). “16. Jacobian Elliptic Functions and Theta Functions”. In Abramowitz, Milton; Stegun, Irene Ann (eds.). Handbook of Mathematical Functions. National Bureau of Standards. pp. 567–585.
Neuman, Edward (2007). “On Gauss lemniscate functions and lemniscatic mean” (PDF). Mathematica Pannonica. 18 (1): 77–94.
Nishimura, Ryo (2015). “New properties of the lemniscate function and its transformation”. Journal of Mathematical Analysis and Applications. 427 (1): 460–468. doi:10.1016/j.jmaa.2015.02.066.
Ogawa, Takuma (2005). “Similarities between the trigonometric function and the lemniscate function from arithmetic view point”. Tsukuba Journal of Mathematics. 29 (1).
Peirce, Charles Sanders (1879). “A Quincuncial Projection of the Sphere”. American Journal of Mathematics. 2 (4): 394–397. doi:10.2307/2369491.
Popescu-Pampu, Patrick (2016). What is the Genus?. Lecture Notes in Mathematics. Vol. 2162. Springer. doi:10.1007/978-3-319-42312-8.
Prasolov, Viktor; Solovyev, Yuri (1997). “4. Abel’s Theorem on Division of Lemniscate”. Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170.
Rančić, Miodrag; Purser, R. James; Mesinger, Fedor (1996). “A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates”. Quarterly Journal of the Royal Meteorological Society. 122 (532): 959–982. doi:10.1002/qj.49712253209.
Reinhardt, William P.; Walker, Peter L. (2010a). “22. Jacobian Elliptic Functions”. In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Reinhardt, William P.; Walker, Peter L. (2010b). “23. Weierstrass Elliptic and Modular Functions”. In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Robinson, Paul L. (2019a). “The Lemniscatic Functions”. arXiv:1902.08614.
Robinson, Paul L. (2019b). “The Elliptic Functions in a First-Order System”. arXiv:1903.07147.
Rosen, Michael (1981). “Abel’s Theorem on the Lemniscate”. The American Mathematical Monthly. 88 (6): 387–395. doi:10.2307/2321821.
Roy, Ranjan (2017). Elliptic and Modular Functions from Gauss to Dedekind to Hecke. Cambridge University Press. p. 28. ISBN 978-1-107-15938-9.
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.
Schneider, Theodor (1937). “Arithmetische Untersuchungen elliptischer Integrale” [Arithmetic investigations of elliptic integrals]. Mathematische Annalen (in German). 113 (1): 1–13. doi:10.1007/BF01571618.
Schwarz, Hermann Amandus (1869). “Ueber einige Abbildungsaufgaben” [About some mapping problems]. Crelle’s Journal (in German). 70: 105–120. doi:10.1515/crll.1869.70.105.
Siegel, Carl Ludwig (1969). “1. Elliptic Functions”. Topics in Complex Function Theory, Vol. I. Wiley-Interscience. pp. 1–89. ISBN 0-471-60844-0.
Snape, Jamie (2004). “Bernoulli’s Lemniscate”. Applications of Elliptic Functions in Classical and Algebraic Geometry (Thesis). University of Durham. pp. 50–56.
Southard, Thomas H. (1964). “18. Weierstrass Elliptic and Related Functions”. In Abramowitz, Milton; Stegun, Irene Ann (eds.). Handbook of Mathematical Functions. National Bureau of Standards. pp. 627–683.
Sridharan, Ramaiyengar (2004) “Physics to Mathematics: from Lintearia to Lemniscate”. Resonance.
“Part I”. 9 (4): 21–29. doi:10.1007/BF02834853.
“Part II: Gauss and Landen’s Work”. 9 (6): 11–20. doi:10.1007/BF02839214.
Todd, John (1975). “The lemniscate constants”. Communications of the ACM. 18 (1): 14–19. doi:10.1145/360569.360580.
Vigren, Erik; Dieckmann, Andreas (21 June 2020). “Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges”. Symmetry. 12 (6): 1040. doi:10.3390/sym12061040.
Whittaker, Edmund Taylor; Watson, George Neville (1920) [1st ed. 1902]. “22.8 The lemniscate functions”. A Course of Modern Analysis (3rd ed.). Cambridge. pp. 524–528.

Lemniscate elliptic functions – Wikipedia

Lemniscate sine and cosine functions[edit]

Definitions[edit]

Arc length of Bernoulli’s lemniscate[edit]

Arc length of rectangular elastica[edit]

Elliptic characterization[edit]

Relation to the lemniscate constant[edit]

Zeros, poles and symmetries[edit]

The lemniscate sine as a ratio of entire functions[edit]

Pythagorean-like identity[edit]

Derivatives and integrals[edit]

Argument sum and multiple identities[edit]

Lemnatomic polynomials[edit]

Specific values[edit]

Power series[edit]

Relation to Weierstrass and Jacobi elliptic functions[edit]

Relation to the modular lambda function[edit]

Ramanujan’s cos/cosh identity[edit]

Continued fractions[edit]

Methods of computation[edit]

Inverse functions[edit]

Expression using elliptic integrals[edit]

Use in integration[edit]

Hyperbolic lemniscate functions[edit]

Number theory[edit]

Hurwitz numbers[edit]

Appearances in Laurent series[edit]

World map projections[edit]

See also[edit]

External links[edit]

References[edit]

Recent Posts

Recent Comments

Archives

Categories

Meta