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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[[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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=== 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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[[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>\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 ====