The Lemniscate Sine — The Lemniscate Sine

2 The Lemniscate Sine

The lemniscate sine is an example of an elliptic integral. Elliptic integrals first arose when people studied problems analogous to the above but for ellipses, and integrals that looked similar were called elliptic integrals. We shall not be interested in ellipses here, because they are less interesting. Fix two foci F±=(±a,0)R2F_{\pm } = (\pm a, 0) \in \mathbb {R}^2, and consider the locus of all points P=(x,y)P = (x, y) such that

PF+PF=0F+0F. \| P - F_+\| \cdot \| P - F_-\| = \| 0 - F_+\| \cdot \| 0 - F_-\| .
\begin{tikzpicture} 
    \draw [blue, semithick, domain=-40:40, variable=\t] plot ({2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * cos (\t)}, {2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * sin (\t)});
    \draw [blue, semithick, domain=45:40, variable=\t, samples=50] plot ({2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * cos (\t)}, {2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * sin (\t)});
    \draw [blue, semithick, domain=-45:-40, variable=\t, samples=50] plot ({2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * cos (\t)}, {2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * sin (\t)});

    \draw [blue, semithick, domain=140:220, variable=\t] plot ({2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * cos (\t)}, {2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * sin (\t)});
    \draw [blue, semithick, domain=135:140, variable=\t, samples=50] plot ({2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * cos (\t)}, {2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * sin (\t)});
    \draw [blue, semithick, domain=225:220, variable=\t, samples=50] plot ({2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * cos (\t)}, {2.5 * sqrt(sqrt(abs(2 * cos (2 * \t)))) * sin (\t)});
    \draw (-4, 0) -- (4, 0) node [right] {$x$};
    \draw (0, -2) -- (0, 2) node [above] {$y$};
    \draw (0, 0) -- (2.1651, 1.25) node [pos=0.5, above] {$r$};

    \draw (0.5, 0) arc(0:30:0.5) node [pos=0.7, right] {$\theta$};
    \node [fill, circle, inner sep = 0, minimum size = 3] at (2.1, 0) {};
    \node [below] at (2.1, 0) {$a$};

    \node [fill, circle, inner sep = 0, minimum size = 3] at (-2.1, 0) {};
    \node [below] at (-2.1, 0) {$-a$};
  \end{tikzpicture}

Explicitly, this is given by

((xa)2+y2)((x+a)2+y2)=a4. ((x - a)^2 + y^2)((x + a)^2 + y^2) = a^4.

Equivalently, this is

(x2+y2)2=2a2(x2y2). (x^2 + y^2)^2 = 2a^2 (x^2 - y^2).

It is convenient to set a=12a = \frac{1}{\sqrt{2}}, and using polar coordinates, this equation becomes

r2=cos2θ. r^2 = \cos 2\theta .

We then have

dθ=rsin2θ  dr=r1r4  dr. \mathrm{d}\theta = \frac{-r}{\sin 2\theta }\; \mathrm{d}r = \frac{-r}{\sqrt{1 - r^4}}\; \mathrm{d}r.

Thus, the line element is

ds=11r4  dr, \mathrm{d}s = \frac{1}{\sqrt{1 - r^4}}\; \mathrm{d}r,

and we have a lemniscate arcsine

s(r0)=0r011r4  dr s(r_0) = \int _0^{r_0} \frac{1}{\sqrt{1 - r^4}}\; \mathrm{d}r

with inverse sl(s)\operatorname{sl}(s). Again, we want to promote this to a complex function. From now on, we will replace rr with xx, since explicit coordinates for the lemniscate itself will not be of much use. Then the associated Riemann surface is given by

R={(x,y)C2:y2=1x4}. R = \{ (x, y) \in \mathbb {C}^2: y^2 = 1 - x^4\} .

Then

s=dxy. s = \int \frac{\mathrm{d}x}{y}.

Observe that this surface is singular at infinity. Which is bad. However, we can blow this up at infinity to resolve the singularity, and since we are working with curves, any rational map is automatically a morphism, and so we don't have to think about infinity much.

A key observation is that projection onto the xx coordinate exhibits RR as a double cover of P1\mathbb {P}^1 branched at four points (it cannot be branched at infinity since there is always an even number of branch points), and so by Riemann-Hurwitz, RR is a torus. This means RR admits a holomorphic and non-vanishing differential, and one can check dxy\frac{\mathrm{d}x}{y} is one.

Since RR is topologically a torus, its homology is generated by two loops, and ss will be defined up to the integrals of those loops, which we expect to be an integer lattice Λ\Lambda in C\mathbb {C}. We will explicitly identify this lattice soon. What this means is that ss actually gives an explicit identification RC/ΛR \overset {\sim }{\to } \mathbb {C}/\Lambda . Surjectivity is automatic, and injectivity is given by the Abel–Jacobi theorem, since if s(p)=s(p)s(p) = s(p'), then ppp - p' is killed by the Abel–Jacobi map, and hence ppp \sim p'. This is possible only if p=pp = p' as RR is not P1\mathbb {P}^1.

To understand the lattice, we pick two generated loops as follows:

\begin{tikzpicture} 
    \draw [gray] (-2, 0) -- (2, 0) node [right] {$\mathrm{Re}$};
    \draw [gray] (0, -2) -- (0, 2) node [above] {$\mathrm{Im}$};

    \node at (0, 1.3) {$\times$};
    \node at (0, -1.3) {$\times$};
    \node at (1.3, 0) {$\times$};
    \node at (-1.3, 0) {$\times$};

    \node [circ] at (0, 0) {};

    \draw [red, semithick] (0, -0.1) -- (1.3, -0.1) arc(-90:90:0.1) -- (-1.3, 0.1) arc(90:270:0.1) -- (0, -0.1);

    \draw [blue, semithick] (0, -0.2) -- (1.3, -0.2) arc(-90:45:0.2) -- (0.1414, 1.4414) arc(45:180:0.2) -- (-0.2, 0) arc(180:270:0.2);
  \end{tikzpicture}

The horizontal loop has a very geometric interpretation, since the path lies entirely within the real axis. Indeed, this just corresponds to integrating the line element along the whole lemniscate, and the integral is just the total arc length of the lemniscate, 2Ω2\Omega . It is written this way so that the length of half of the lemniscate is Ω\Omega , which is exactly the contribution we get if we go around the branch point at 11 and then return to the “origin” (of course, on the Riemann surface, this is a distinct point that the origin, as it is on the other sheet).

It remains to understand what happens when we go around the branch point at ii. This follows from the simple observation that the integrand is invariant under xixx \mapsto ix, while dx\mathrm{d}x gets multiplied by ii under this operation. So we find that

s(iz)=is(z). s(iz) = i s(z).

So it follows that the other homology class has an integral of (1+i)Ω(1 + i)\Omega . Thus,

Λ=2Ω,(1+i)Ω. \Lambda = \langle 2\Omega , (1 + i)\Omega \rangle .

The inverse function sl\operatorname{sl} is then a double periodic function with period lattice Λ\Lambda with zeroes given by Λ(Ω+Λ)\Lambda \cup (\Omega + \Lambda ).