Lemniscate elliptic functions: Difference between revisions

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The lemniscate functions have periods related to a number {{math|<math>\varpi =</math> 2.622057...}} called the [[lemniscate constant]], the ratio of a lemniscate's perimeter to its diameter. This number is a [[Quartic plane curve|quartic]] analog of the ([[Conic section|quadratic]]) {{math|<math>\pi =</math> 3.141592...}}, [[pi|ratio of perimeter to diameter of a circle]].

As [[complex analysis|complex functions]], {{math|sl}} and {{math|cl}} have a [[square lattice|square]] [[period lattice]] (a multiple of the [[Gaussian integer]]s) with [[Fundamental pair of periods|fundamental periods]] <math>\{(1 + i)\varpi, (1 - i)\varpi\},</math><ref>The fundamental periods <math>(1+i)\varpi</math> and <math>(1-i)\varpi</math> are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative.</ref> and are a special case of two [[Jacobi elliptic functions]] on that lattice, <math>\operatorname{sl} z = \operatorname{sn}(z; i-1),</math> <math>\operatorname{cl} z = \operatorname{cd}(z; i-1)</math>.

Similarly, the '''hyperbolic lemniscate sine''' {{math|slh}} and '''hyperbolic lemniscate cosine''' {{math|clh}} have a square period lattice with fundamental periods <math>\bigl\{\sqrt2\varpi, \sqrt2\varpi i\bigr\}.</math>

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:<math> 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}}.</math>

Beyond that square, the functions can be [[analytic continuation|analytically continued]]extended to the whole [[complex plane]] byvia [[analytic continuation]] aby series ofsuccessive [[Schwarz reflection principle|reflections]].

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

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[[File:Lemniscate constant as an integral.png|thumb|upright=1.3|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 {{mvar|ϖ}}.]]

The lemniscate functions have minimal real period {{mathtmath|2''ϖ''\varpi}}, minimal [[Imaginary number|imaginary]] period {{mathtmath|2''ϖ''\varpi i}} and fundamental complex periods <math>(1+i)\varpi</math> and <math>(1-i)\varpi</math> for a constant {{mathtmath|''ϖ''\varpi}} called the ''[[lemniscate constant]]'',<ref>{{harvp|Schappacher|1997}}. OEIS sequence [https://oeis.org/A062539 A062539] lists the lemniscate constant's decimal digits.</ref>

:<math>\varpi = 2\int_0^1\frac{\mathrm{d}t}{\sqrt{1-t^4}} = 2.62205\ldots</math>

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The lemniscate functions satisfy the basic relation <math>\operatorname{cl}z = {\operatorname{sl}}\bigl(\tfrac12\varpi - z\bigr),</math> analogous to the relation <math>\cos z = {\sin}\bigl(\tfrac12\pi - z\bigr).</math>

The lemniscate constant {{mathtmath|''ϖ''\varpi}} is a close analog of the [[pi|circle constant {{mathtmath|''π''\pi}}]], and many identities involving {{mathtmath|''π''\pi}} have analogues involving {{mathtmath|''ϖ''\varpi}}, as identities involving the [[trigonometric functions]] have analogues involving the lemniscate functions. For example, [[Viète's formula]] for {{mathtmath|''π''\pi}} can be written:

<math display="block">

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</math>

An analogous formula for {{mathtmath|''ϖ''\varpi}} is:<ref>{{harvp|Levin|2006}}</ref>

<math display="block">

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</math>

The [[Machin-like formula|Machin formula]] for {{mathtmath|''π''\pi}} is <math display="inline">\tfrac14\pi = 4 \arctan \tfrac15 - \arctan \tfrac1{239},</math> and several similar formulas for {{mathtmath|''π''\pi}} can be developed using trigonometric angle sum identities, e.g. Euler's formula <math display="inline">\tfrac14\pi = \arctan\tfrac12 + \arctan\tfrac13</math>. Analogous formulas can be developed for {{mathtmath|''ϖ''\varpi}}, including the following found by Gauss: <math>\tfrac12\varpi = 2 \operatorname{arcsl} \tfrac12 + \operatorname{arcsl} \tfrac7{23}.</math><ref>{{harvp|Todd|1975}}</ref>

The lemniscate and circle constants were found by Gauss to be related to each-other by the [[arithmetic-geometric mean]] {{mathtmath|''M''}}:<ref>{{harvp|Cox|1984}}</ref>

<math display="block">

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\end{aligned}</math>

As a result, both functions are invariant under translation by an [[Gaussian integer#Examples|even-Gaussian-integer]] multiple of <math>\varpi</math>.<ref>The even Gaussian integers are the residue class of {{tmath|0}}, modulo {{mathtmath|1 + ''i''}}, the black squares on a [[Checkerboard#Mathematical description|checkerboard]].</ref> That is, a displacement <math>(a + bi)\varpi,</math> with <math>a + b = 2k</math> for integers {{mathtmath|''a''}}, {{mathtmath|''b''}}, and {{mathtmath|''k''}}.

:<math>\begin{aligned}

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\end{aligned}</math>

The {{math|sl}} function has simple [[zeros and poles|zeros]] at Gaussian integer multiples of {{mathtmath|''ϖ''\varpi}}, complex numbers of the form <math>a\varpi + b\varpi i</math> for integers {{mathtmath|''a''}} and {{mathtmath|''b''}}. It has simple [[zeros and poles|poles]] at Gaussian [[half-integer]] multiples of {{mathtmath|''ϖ''\varpi}}, complex numbers of the form <math>\bigl(a + \tfrac12\bigr)\varpi + \bigl(b + \tfrac12\bigr)\varpi i</math>, with [[residue (complex analysis)|residue]]s <math>(-1)^{a-b+1}i</math>. The {{math|cl}} function is reflected and offset from the {{math|sl}} function, <math>\operatorname{cl}z = {\operatorname{sl}}\bigl(\tfrac12\varpi - z\bigr)</math>. It has zeros for arguments <math>\bigl(a + \tfrac12\bigr)\varpi + b\varpi i</math> and poles for arguments <math>a\varpi + \bigl(b + \tfrac12\bigr)\varpi i,</math> with residues <math>(-1)^{a-b}i.</math>

Also

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:<math>

\operatorname{cl}^2 \tfrac12x = \frac{1+\operatorname{cl}x \sqrt{1+\operatorname{sl}^2x}}{1 + \sqrt{1+\operatorname{sl}^2x}+1}

</math>

:<math>

\operatorname{sl}^2 \tfrac12x = \frac{1-\operatorname{cl}x\sqrt{1+\operatorname{sl}^2x}}{1 + \sqrt{1+\operatorname{sl}^2x}+1}

</math>

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===Specific values===

Just as for the trigonometric functions, values of the lemniscate functions can be computed for divisions of the lemniscate into {{mathtmath|''n''}} parts of equal length, using only basic arithmetic and square roots, if and only if {{mathtmath|''n''}} is of the form <math>n = 2^kp_1p_2\cdots p_m</math> where {{mathtmath|''k''}} is a non-negative [[integer]] and each {{mathtmath|''p''_''i''p_i}} (if any) is a distinct [[Fermat prime]].<ref>{{harvp|Rosen|1981}}</ref>

{| class="wikitable"

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[[File:Lemniscate 15-gon.png|thumb|right|upright=1.5|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 polygon]]s in the circle, the lemniscate can be divided into {{mathtmath|''n''}} sections of equal arc length using only [[straightedge and compass construction|straightedge and compass]] if and only if {{mathtmath|''n''}} is of the form <math>n = 2^kp_1p_2\cdots p_m</math> where {{mathtmath|''k''}} is a non-negative [[integer]] and each {{mathtmath|''p''_''i''p_i}} (if any) is a distinct [[Fermat prime]].<ref>Such numbers are OEIS sequence [[oeis:A003401|A003401]].</ref> The "if" part of the theorem was proved by [[Niels Henrik Abel|Niels Abel]] in 1827–1828, and the "only if" part was proved by [[Michael Rosen (mathematician)|Michael Rosen]] in 1981.<ref>{{harvp|Abel|1827–1828}}; {{harvp|Rosen|1981}}; {{harvp|Prasolov|Solovyev|1997}}</ref> Equivalently, the lemniscate can be divided into {{mathtmath|''n''}} sections of equal arc length using only straightedge and compass if and only if <math>\varphi (n)</math> is a [[power of two]] (where <math>\varphi</math> is [[Euler's totient function]]). The lemniscate is ''not'' assumed to be already drawn, as that would go against the rules of straightedge and compass constructions; instead, it is assumed that we are given only two points by which the lemniscate is defined, such as its center and radial point (one of the two points on the lemniscate such that their distance from the center is maximal) or its two foci.

Let <math>r_j=\operatorname{sl}\dfrac{2j\varpi}{n}</math>. Then the {{mathtmath|''n''}}-division points for <math>\mathcal{L}</math> are the points

:<math>\left(r_j\sqrt{\tfrac12\bigl(1+r_j^2\bigr)},\ (-1)^{\left\lfloor 4j/n\right\rfloor} \sqrt{\tfrac12r_j^2\bigl(1-r_j^2\bigr)}\right),\quad j\in\{1,2,\ldots ,n\}</math>

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=== Arc length of rectangular elastica ===

[[File:Rectangular elastica and lemniscatic sine.png|thumb|upright=1.3|The lemniscate sine relates the arc length to the x coordinate in the rectangular elastica.]]

The inverse lemniscate sine also describes the arc length {{mathtmath|''s''}} relative to the {{mathtmath|''x''}} coordinate of the rectangular [[Elastica theory|elastica]].<ref>{{harvp|Euler|1786}}; {{harvp|Sridharan|2004}}; {{harvp|Levien|2008}}</ref> This curve has {{mathtmath|''y''}} coordinate and arc length:

:<math>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}}</math>

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}}

| <math>\phi_N \leftarrow 2^N a_N \sqrt2x</math>

| '''for each''' {{mathtmath|''n''}} from {{mathtmath|''N''}} to {{mathtmath|0}} '''do'''

{{ubl |item_style=padding:0.2em 0 0 1.6em;

| <math>\phi_{n-1} \leftarrow \tfrac12\left(\phi_n + {\arcsin}{\left(\frac{c_n}{a_n}\sin \phi_n\right)}\right)</math>

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:<math>M(z)=z\exp\left(-\int_0^z\int_0^w \left(\frac{1}{\operatorname{sl}^2v}-\frac{1}{v^2}\right)\, \mathrm dv\,\mathrm dw\right),</math>

:<math>N(z)=\exp\left(\int_0^z\int_0^w \operatorname{sl}^2v\,\mathrm dv\,\mathrm dw\right)</math>

where the contours do not cross the poles; while the innermost integrals are path-independent, the outermost ones are path-dependent; however, the path dependence cancels out with the non-injectivity of the complex [[exponential function]].

An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see [[#Methods of computation|Lemniscate elliptic functions § Methods of computation]]); the relation between <math>M,N</math> and <math>\theta_1,\theta_3</math> is

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=== Relation to Weierstrass and Jacobi elliptic functions ===

The lemniscate functions are closely related to the [[Weierstrass elliptic function]] <math>\wp(z; 1, 0)</math> (the "lemniscatic case"), with invariants {{mathtmath|''g''₂1= g_2 {{=}} 1}} and {{mathtmath|''g''₃1= g_3 {{=}} 0}}. This lattice has fundamental periods <math>\omega_1 = \sqrt{2}\varpi,</math> and <math>\omega_2 = i\omega_1</math>. The associated constants of the Weierstrass function are <math>e_1=\tfrac12,\ e_2=0,\ e_3=-\tfrac12.</math>

The related case of a Weierstrass elliptic function with {{mathtmath|''g''₂1= g_2 {{=}} ''a''}}, {{mathtmath|''g''₃1= 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: {{mathtmath|''a'' > 0}} and {{mathtmath|''a'' < 0}}. The period [[parallelogram]] is either a [[square]] or a [[rhombus]]. The Weierstrass elliptic function <math>\wp (z;-1,0)</math> is called the "pseudolemniscatic case".<ref>{{harvp|Robinson|2019a}}</ref>

The square of the lemniscate sine can be represented as

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:<math>\operatorname{sl}^2 z=\frac{1}{\wp (z;4,0)}=\frac{i}{2\wp ((1-i)z;-1,0)}={-2\wp}{\left(\sqrt2z+(i-1)\frac{\varpi}{\sqrt2};1,0\right)}</math>

where the second and third argument of <math>\wp</math> denote the lattice invariants {{mathtmath|''g''₂g_2}} and {{mathtmath|''g''₃g_3}}. The lemniscate sine is a [[rational function]] in the Weierstrass elliptic function and its derivative:<ref>{{harvp|Eymard|Lafon|2004}} p. 234</ref>

:<math>\operatorname{sl}z=-2\frac{\wp (z;-1,0)}{\wp '(z;-1,0)}.</math>

The lemniscate functions can also be written in terms of [[Jacobi elliptic functions]]. The Jacobi elliptic functions <math>\operatorname{sn}</math> and <math>\operatorname{cd}</math> with positive real elliptic modulus have an "upright" rectangular lattice aligned with real and imaginary axes. Alternately, the functions <math>\operatorname{sn}</math> and <math>\operatorname{cd}</math> with modulus {{mathtmath|''i''}} (and <math>\operatorname{sd}</math> and <math>\operatorname{cn}</math> with modulus <math>1/\sqrt{2}</math>) have a square period lattice rotated 1/8 turn.<ref>{{Cite book |last1=Armitage |first1=J. V. |title=Elliptic Functions |last2=Eberlein |first2=W. F. |publisher=Cambridge University Press |year=2006 |isbn=978-0-521-78563-1 |page=49}}</ref><ref>The identity <math>\operatorname{cl} z = {\operatorname{cn}}\left(\sqrt2z;\tfrac{1}{\sqrt2}\right)</math> can be found in {{harvp|Greenhill|1892}} [[iarchive:applicationselli00greerich/page/n48|p. 33]].</ref>

:<math> \operatorname{sl} z = \operatorname{sn}(z;i)=\operatorname{sc}(z;\sqrt{2})={\tfrac1{\sqrt2}\operatorname{sd}}\left(\sqrt2z;\tfrac{1}{\sqrt2}\right) </math>

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&\quad {}= \frac 2 {2 + \sqrt{7} + \sqrt{21 + 8 \sqrt{7}} + \sqrt{2 {14 + 6 \sqrt{7} + \sqrt{455 + 172 \sqrt{7}}}}}

\\[18mu]

& {\operatorname{sl}}\bigl(\tfrac1{18}\varpi\bigr)\, {\operatorname{sl}}\bigl(\tfrac3{18}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac5{18}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac7{18}\varpi\bigr) \\[-3mu]

&\quad {}= \sqrt[8]{\frac{\lambda (9i)}{1-\lambda (9i)}}

= {\tan}\Bigglleft(\vphantom{\frac\Big|\Big|}\right. \frac\pi4 - {\arctan}\Bigglleft(\vphantom{\frac\Big|\Big|}\right.\frac{2\sqrt[3]{2\sqrt{3}-2}-2\sqrt[3]{2-\sqrt{3}}+\sqrt{3}-1}{\sqrt[4]{12}}\Biggrleft.\left.\vphantom{\frac\Big|\Big|}\right)\Biggrright)

\end{aligned}</math>

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:<math> \operatorname{arccl} x = \int_{x}^{1} \frac{\mathrm dt}{\sqrt{1-t^4}} = \tfrac12\varpi - \operatorname{arcsl}x</math>

For {{mathtmath|''x''}} in the interval <math>-1 \leq x \leq 1</math>, <math>\operatorname{sl}\operatorname{arcsl} x = x</math> and <math>\operatorname{cl}\operatorname{arccl} x = x</math>

For the halving of the lemniscate arc length these formulas are valid:{{cn|date=September 2024}}

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=== Use in integration ===

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

:<math>\int\frac{1}{\sqrt{1-x^4}}\,\mathrm dx=\operatorname{arcsl} x</math>

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[[File:Superellipse chamfered square.svg|thumb|280x280px|Superellipse with the relation <math>x^4 + y^4 = 1</math>]]

In a quartic [[Fermat curve]] <math>x^4 + y^4 = 1</math> (sometimes called a [[squircle]]) the hyperbolic lemniscate sine and cosine are analogous to the tangent and cotangent functions in a unit circle <math>x^2 + y^2 = 1</math> (the quadratic Fermat curve). If the origin and a point on the curve are connected to each other by a line {{mathtmath|''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 {{mathtmath|''L''}} with the line <math>x = 1</math>.<ref>{{harvp|Levin|2006}}; {{harvp|Robinson|2019b}}</ref> Just as <math>\pi</math> is the area enclosed by the circle <math>x^2+y^2=1</math>, the area enclosed by the squircle <math>x^4+y^4=1</math> is <math>\sigma</math>. Moreover,

:<math>M(1,1/\sqrt{2})=\frac{\pi}{\sigma}</math>

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:<math> \operatorname{slh}(a+b) = \frac{\operatorname{slh}a\operatorname{slh}'b + \operatorname{slh}b\operatorname{slh}'a}{1-\operatorname{slh}^2a\,\operatorname{slh}^2b} </math>

When <math>u</math> is real, the derivative and the original [[antiderivative]] of <math> \operatorname{slh} </math> and <math> \operatorname{clh} </math> can be expressed in this way:

:{|class = "wikitable"

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:<math>\text{ctlh}(u) = \frac{\text{cd}(u;\tfrac{1}{2}\sqrt{2})}{\sqrt[4]{\text{cd}(u;\tfrac{1}{2}\sqrt{2})^4 + \text{sn}(u;\tfrac{1}{2}\sqrt{2})^4}} </math>

When <math>u</math> is real, the derivative and [[quarter period]] integral of <math> \operatorname{tlh} </math> and <math> \operatorname{ctlh} </math> can be expressed in this way:

:{|class = "wikitable"

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:<math>y(w) = \text{tlh}(w) </math>

The sketch also shows the fact that the derivation of the Areasinus hyperbolic lemniscatus function is equal to the reciprocal of the square root of the successor of the [[fourth power]] function.

==== First proof: comparison with the derivative of the arctangent ====

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:<math>\mathrm{H}_8=\frac{3}{10},\,\mathrm{H}_{12}=\frac{567}{130},\,\mathrm{H}_{16}=\frac{43\,659}{170},\,\ldots</math>

Also<ref>{{cite journal |last1=Katz |first1=Nicholas M. |date=1975 |title=The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers |url=https://link.springer.com/article/10.1007/BF02547966 |journal=Mathematische Annalen |volume=216 |issue=1 |pages=1–4|doi=10.1007/BF02547966 }} See eq. (9)</ref>

:<math>\operatorname{denom}\mathrm{H}_{4n}=\prod_{(p-1)|4n}p</math>

where <math>p\in\mathbb{P}</math> such that <math>p\not\equiv 3\,(\text{mod}\,4),</math>

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If <math>a</math> and <math>p</math> are coprime, then there exist numbers <math>p'\in\mathbb{Z}[i]</math> (see<ref>{{cite journal |last1=Eisenstein |first1=G.

|title=Beiträge zur Theorie der elliptischen Functionen |language=German|journal=Journal für die reine und angewandte Mathematik|date=1846 |volume=30| url=https://gdz.sub.uni-goettingen.de/id/PPN243919689_0030?tify=%7B%22pages%22%3A%5B202%5D%2C%22view%22%3A%22scan%22%7D}} Eisenstein uses <math>\varphi=\operatorname{sl}</math> and <math>\omega=2\varpi</math>.</ref> for these numbers) such that<ref>{{cite journal harvp|last1=Ogawa |first1=Takuma |title=Similarities between the trigonometric function and the lemniscate function from arithmetic view point |journal=Tsukuba Journal of Mathematics |date=2005 |volume=29 |issue=1 |doi=10.21099/tkbjm/1496164894 |url=https://projecteuclid.org/journals/tsukuba-journal-of-mathematics/volume-29/issue-1/Similarities-between-the-trigonometric-function-and-the-lemniscate-function-from/10.21099/tkbjm/1496164894.full }}</ref>

:<math>\left(\frac{a}{p}\right)_4=\prod_{p'} \frac{\operatorname{sl}(2\varpi ap'/p)}{\operatorname{sl}(2\varpi p'/p)}.</math>

This theorem is analogous to

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== Notes ==

{{Reflist|30em}}

== External links ==

* {{Cite episode |last1=Parker |first1=Matt |author-link1=Matt Parker |title=What is the area of a Squircle?|url=https://www.youtube.com/watch?v=gjtTcyWL0NA |archive-url=https://ghostarchive.org/varchive/youtube/20211219/gjtTcyWL0NA |archive-date=2021-12-19 |url-status=live|series=Stand-up Maths |date=2021 |network=YouTube }}{{cbignore}}

* {{Cite episode |last=Ramalingam |first=Muthu Veerappan|title=Bernoulli Lemniscate and the Squircle <math>||</math> A remarkable Geometric fun fact!!? |url=https://www.youtube.com/watch?v=mAzIE5OkqWE |archive-url=https://ghostarchive.org/varchive/mAzIE5OkqWE |archive-date=2024-11-10 |url-status=live|series=Act of Learning |date=2023 |network=YouTube}}{{cbignore}} Relation shown in the video amounts to <math>\operatorname{cl}(\sqrt{2}t)=\frac{\cos_4^2(t)-\sin_4^2(t)}{\cos_4^2(t)+\sin_4^2(t)}</math>

== References ==

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* {{Cite book |last1=Berndt |first1=Bruce C. |title=Ramanujan's Notebooks Part IV |publisher=Springer |year=1994 |edition=First |isbn=978-1-4612-6932-8 }}

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* {{cite book |last1=Bottazzini |first1=Umberto |authorlink1=Umberto Bottazzini |last2=Gray |first2=Jeremy |authorlink2=Jeremy Gray (mathematician) |date=2013 |title=Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory |publisher=Springer |doi=10.1007/978-1-4614-5725-1 |isbn=978-1-4614-5724-4 }}

* {{cite book |last=Carlson |first=Billie C. |year=2010 |authorlink1=Bille C. Carlson |display-editors=1 |editor1-last=Olver |editor1-first=Frank |editor1-link=Frank W. J. Olver |editor2-last=Lozier |editor2-first=Daniel |editor3-last=Boisvert |editor3-first=Ronald |editor4-last=Clark |editor4-first=Charles |title=NIST Handbook of Mathematical Functions |publisher=Cambridge |chapter=19. Elliptic Integrals |chapter-url=https://dlmf.nist.gov/19 |title-link=Digital Library of Mathematical Functions }}

* {{cite journal |last=Cox |first=David Archibald |authorlink1=David A. Cox |date=January 1984 |title=The Arithmetic-Geometric Mean of Gauss |url=https://www.researchgate.net/publication/248675540 |journal=L'Enseignement Mathématique |volume=30 |issue=2 |pages=275–330 }}

* {{cite journal |last1=Cox |first1=David Archibald |last2=Shurman |first2=Jerry |date=2005 |title=Geometry and number theory on clovers |journal=The American Mathematical Monthly |volume=112 |issue=8 |pages=682–704 |doi=10.1080/00029890.2005.11920241 |url=https://people.reed.edu/~jerry/Clover/cloverexcerpt.pdf }}

* {{cite book |last1=Cox |first1=David Archibald |year=2012 |chapter=The Lemniscate |pages=463–514 |title=Galois Theory |publisher=Wiley |doi=10.1002/9781118218457.ch15 |isbn=978-1-118-07205-9 }}

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* {{cite book |last1= Houzel |first1=Christian |author-link1=Christian Houzel |date=1978 |chapter=Fonctions elliptiques et intégrales abéliennes |trans-chapter=Elliptic functions and Abelian integrals |editor-last=Dieudonné |editor-first= Jean |editor-link1=Jean Dieudonné |title= Abrégé d'histoire des mathématiques, 1700–1900. II |publisher=Hermann |pages=1–113 |language=fr}}

* {{cite journal |last1=Hyde |first1=Trevor |year=2014 |title=A Wallis product on clovers |journal=The American Mathematical Monthly |volume=121 |issue=3 |pages=237–243 |url=https://math.uchicago.edu/~tghyde/Hyde%20--%20A%20Wallis%20product%20on%20clovers.pdf |doi=10.4169/amer.math.monthly.121.03.237 }}

* {{cite journal |last1=Kubota |first1=Tomio |authorlink1=Tomio Kubota |date=1964 |title=Some arithmetical applications of an elliptic function |journal=Crelle's Journal |volume='''214'''/'''215''' |pages=141–145 |doi=10.1515/crll.1964.214-215.141 }}

* {{cite journal |last1=Langer |first1=Joel C. |first2=David A. |last2=Singer |year=2010 |title=Reflections on the Lemniscate of Bernoulli: The Forty-Eight Faces of a Mathematical Gem |journal=Milan Journal of Mathematics |volume=78 |issue=2 |pages=643–682 |url=https://case.edu/artsci/math/langer/jlpreprints/fortyeight2010.pdf |doi=10.1007/s00032-010-0124-5 }}

* {{cite journal |last1=Langer |first1=Joel C. |first2=David A. |last2=Singer |year=2011 |title=The lemniscatic chessboard |journal=Forum Geometricorum |volume=11 |pages=183–199 |url=https://forumgeom.fau.edu/FG2011volume11/FG201119index.html }}

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| title = Conformal Projections Based on Elliptic Functions

| ___location = Toronto | publisher = B. V. Gutsell, York University

| series = ''Cartographica Monographs'' | volume = 16

| url = https://archive.org/details/conformalproject0000leel | url-access = limited

| isbn = 0-919870-16-3

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* {{Cite book |last1=Lemmermeyer |first1=Franz |title=Reciprocity Laws: From Euler to Eisenstein |publisher=Springer |year=2000 |isbn=3-540-66957-4}}

* {{cite tech report |last1=Levien |first1=Raph |year=2008 |authorlink1=Raph Levien |title=The elastica: a mathematical history |number=UCB/EECS-2008-103 |institution=University of California at Berkeley |url=https://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.pdf}}

* {{cite journal |last1=Levin |first1=Aaron |title=A Geometric Interpretation of an Infinite Product for the Lemniscate Constant |journal=The American Mathematical Monthly |date=2006 |volume=113 |issue=6 |pages=510–520 |doi=10.23071080/2764197600029890.2006.11920331 |jstor=27641976 |url=https://web.archive.org/web/20041220213524id_/http://math.berkeley.edu:80/~adlevin/Lemniscate.pdf }}

* {{cite journal |last1=Lindqvist |first1=Peter |last2=Peetre |first2=Jaak |authorlink2=Jaak Peetre |title=Two Remarkable Identities, Called Twos, for Inverses to Some Abelian Integrals |journal=The American Mathematical Monthly |date=2001 |volume=108 |issue=5 |pages=403–410 |doi=10.1080/00029890.2001.11919766 |url=https://people.math.osu.edu/lang.162/book/LiPe3.pdf |archive-url=https://web.archive.org/web/20220528124044/https://people.math.osu.edu/lang.162/book/LiPe3.pdf |archive-date=28 May 2022-05-28 }}

* {{cite book |last1=Markushevich |first1=Aleksei Ivanovich |authorlink1=Aleksei Ivanovich Markushevich |title=The Remarkable Sine Functions |date= 1966 |publisher=Elsevier |url=https://archive.org/details/markushevich-the-remarkable-sine-functions/ }}

* {{cite book |last1=Markushevich |first1=Aleksei Ivanovich |title=Introduction to the Classical Theory of Abelian Functions |series=Translations of Mathematical Monographs |date=1992 |volume=96 |publisher=American Mathematical Society |doi=10.1090/mmono/096 |isbn=978-0-8218-4164-8 }}

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* {{cite journal |last1=Neuman |first1=Edward |authorlink1=Edward Neuman |year=2007 |title=On Gauss lemniscate functions and lemniscatic mean |journal=Mathematica Pannonica |volume=18 |issue=1 |pages=77–94 |url=http://mathematica-pannonica.ttk.pte.hu/articles/mp18-1/MP18-1(2007)pp077-094.pdf }}

* {{cite journal |last1=Nishimura |first1=Ryo |title=New properties of the lemniscate function and its transformation |journal=Journal of Mathematical Analysis and Applications |date=2015 |volume=427 |issue=1 |pages=460–468 |doi=10.1016/j.jmaa.2015.02.066 |doi-access=free }}

* {{cite journal |last1=Ogawa |first1=Takuma |title=Similarities between the trigonometric function and the lemniscate function from arithmetic view point |journal=Tsukuba Journal of Mathematics |date=2005 |volume=29 |issue=1 |doi=10.21099/tkbjm/1496164894 |urldoi-access=https://projecteuclid.org/journals/tsukuba-journal-of-mathematics/volume-29/issue-1/Similarities-between-the-trigonometric-function-and-the-lemniscate-function-from/10.21099/tkbjm/1496164894.fullfree }}

* {{cite journal |last1=Peirce |first1=Charles Sanders |authorlink1=Charles Sanders Peirce |date=1879 |title=A Quincuncial Projection of the Sphere |journal=American Journal of Mathematics |volume=2 |issue=4 |pages=394–397 |doi=10.2307/2369491 |doi-access=free |jstor=2369491 |jstor-access=free |url=https://archive.org/details/sim_american-journal-of-mathematics_1879_2/page/n403/mode/2up}}

* {{cite book |last=Popescu-Pampu |first=Patrick |date=2016 |title=What is the Genus? |series=Lecture Notes in Mathematics |volume=2162 |publisher=Springer |doi=10.1007/978-3-319-42312-8 |isbn=978-3-319-42311-1 }}

* {{cite book |last1=Prasolov |first1=Viktor |last2=Solovyev |first2=Yuri |date=1997 |chapter=4. Abel's Theorem on Division of Lemniscate |title=Elliptic functions and elliptic integrals |series=Translations of Mathematical Monographs |volume=170 |publisher=American Mathematical Society. |doi=10.1090/mmono/170 |isbn=978-0-8218-0587-9 }}

* {{cite journal| last1=Rančić |first1=Miodrag |last2=Purser |first2=R. James | last3=Mesinger |first3=Fedor |date=1996 |title=A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates |journal=Quarterly Journal of the Royal Meteorological Society |volume=122 |issue=532 |pages=959–982 |doi=10.1002/qj.49712253209 |bibcode=<!-- useless bibcode 1996QJRMS.122..959R --> }}

* {{cite book |last1=Reinhardt |first1=William P. |last2=Walker |first2=Peter L. |year=2010a |display-editors=1 |editor1-last=Olver |editor1-first=Frank |editor2-last=Lozier |editor2-first=Daniel |editor3-last=Boisvert |editor3-first=Ronald |editor4-last=Clark |editor4-first=Charles |title=NIST Handbook of Mathematical Functions |publisher=Cambridge |chapter=22. Jacobian Elliptic Functions |chapter-url=https://dlmf.nist.gov/22 |title-link=Digital Library of Mathematical Functions }}

* {{cite book |last1=Reinhardt |first1=William P. |last2=Walker |first2=Peter L. |year=2010b |display-editors=1 |editor1-last=Olver |editor1-first=Frank |editor2-last=Lozier |editor2-first=Daniel |editor3-last=Boisvert |editor3-first=Ronald |editor4-last=Clark |editor4-first=Charles |title=NIST Handbook of Mathematical Functions |publisher=Cambridge |chapter=23. Weierstrass Elliptic and Modular Functions |chapter-url=https://dlmf.nist.gov/23 |title-link=Digital Library of Mathematical Functions }}

* {{citecitation arXiv|mode=cs1 |last1=Robinson |first1=Paul L. |title=The Lemniscatic Functions |date=2019a |eprintarxiv=1902.08614 |classtype=mathPreprint |url=https://arxiv.CVorg/abs/1903.07147 }}

* {{citecitation arXiv|mode=cs1 |last1=Robinson |first1=Paul L. |title= The Elliptic Functions in a First-Order System |date=2019b |eprintarxiv=1903.07147 |classtype=mathPreprint |url=https://arxiv.CVorg/abs/1903.07147}}

* {{cite journal |last1=Rosen |first1=Michael |authorlink1=Michael Rosen (mathematician) |title=Abel's Theorem on the Lemniscate |journal=The American Mathematical Monthly |date=1981 |volume=88 |issue=6 |pages=387–395 |doi=10.23071080/232182100029890.1981.11995279 |jstor=2321821 }}

* {{cite book |last1=Roy |first1=Ranjan |title=Elliptic and Modular Functions from Gauss to Dedekind to Hecke |publisher=Cambridge University Press |page=28 |year=2017 |isbn=978-1-107-15938-9}}

* {{cite book |chapter=Some milestones of lemniscatomy |last1=Schappacher |first1=Norbert |author-link1=Norbert Schappacher | date= 1997 |editor1-last=Sertöz |editor1-first=S. |title=Algebraic Geometry |type=Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey |publisher=Marcel Dekker |pages=257–290 | chapter-url=http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1997_LemniscProvis.pdf}}

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* {{cite book |last1=Whittaker |first1=Edmund Taylor |authorlink1=Edmund T. Whittaker |last2=Watson |first2=George Neville |authorlink2=George N. Watson |date= 1927 |orig-date=4th ed. 1927 |chapter=21 The theta functions |pages=469–470 |edition=4th |title=A Course of Modern Analysis |title-link=A Course of Modern Analysis |publisher= Cambridge }}

{{refend}}

== External links ==

* {{Cite episode |last1=Parker |first1=Matt |author-link1=Matt Parker |title=What is the area of a Squircle?|url=https://www.youtube.com/watch?v=gjtTcyWL0NA |archive-url=https://ghostarchive.org/varchive/youtube/20211219/gjtTcyWL0NA |archive-date=2021-12-19 |url-status=live|series=Stand-up Maths |date=2021 |network=YouTube }}{{cbignore}}

* {{Cite episode |last=Ramalingam |first=Muthu Veerappan|title=Bernoulli Lemniscate and the Squircle <math>||</math> A remarkable Geometric fun fact!!? |url=https://www.youtube.com/watch?v=mAzIE5OkqWE |archive-url=https://ghostarchive.org/varchive/mAzIE5OkqWE |archive-date=2024-11-10 |url-status=live|series=Act of Learning |date=2023 |network=YouTube}}{{cbignore}} Relation shown in the video amounts to <math>\operatorname{cl}(\sqrt{2}t)=\frac{\cos_4^2(t)-\sin_4^2(t)}{\cos_4^2(t)+\sin_4^2(t)}</math>

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