Start with a familiar function, seen in an odd way. The integral

$$\arcsin x=\int_0^x\frac{dt}{\sqrt{1-t^2}}$$

gives the length of an arc of a circle of radius 1, measured from the lowest point, as a function of the height $x$. It is a complicated function. But its inverse, which gives the height as a function of the arc, is the sine, the simplest function in trigonometry. And the sine has a property the integral hides: it is periodic. Since a quarter turn measures

$$\int_0^1\frac{dt}{\sqrt{1-t^2}}=\frac\pi2,$$

the sine repeats every $4\cdot\frac\pi2=2\pi$.

Integrals nobody could compute

In the eighteenth century similar integrals appeared, but with the root of a polynomial of degree 3 or 4 instead of degree 2. They arose in the length of an arc of an ellipse, which is why they were named elliptic integrals; in the length of the lemniscate; and in the motion of the pendulum.

A pendulum of length $L$ swinging with a small amplitude has period $T_0=2\pi\sqrt{L/g}$, as one learns at school. But the formula only holds for small amplitudes. With any amplitude $\theta_0$, the exact period is

$$T=4\sqrt{\frac Lg}\,K(k),\qquad K(k)=\int_0^1\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}},\qquad k=\sin\frac{\theta_0}2.$$

For $\theta_0=90°$, $k=\frac{\sqrt2}2$ and $K\approx1.8541$. So $T\approx7.416\sqrt{L/g}$, against $6.283\sqrt{L/g}$ from the simple formula: the pendulum takes 18% longer. With $k=0$, $K=\frac\pi2$ and we are back to the school formula.

The integral $K$ cannot be written with known functions. The Frenchman Adrien-Marie Legendre devoted forty years to these integrals, classified them into three kinds and computed huge tables of values. His Treatise on elliptic functions appeared between 1825 and 1828.

Abel came to these integrals on the advice of Ferdinand Degen. In 1821, after reading the young Norwegian’s false solution of the quintic, the Copenhagen mathematician recommended that he study them: they would be a “Strait of Magellan” leading to an ocean of discoveries.

Abel’s idea

Abel had the idea Legendre did not: to do with these integrals what one does with the arcsine, that is, to invert them. Instead of studying the arc as a function of the height, study the height as a function of the arc. In the simplest case, define

$$s=\int_0^x\frac{dt}{\sqrt{1-t^4}}\qquad\Longrightarrow\qquad x=\operatorname{sl}(s),$$

the lemniscatic sine. As with the sine, the inverse function is much better behaved than the integral. It goes from 0 to 1 as $s$ goes from 0 to

$$\frac\varpi2=\int_0^1\frac{dt}{\sqrt{1-t^4}}\approx1.31103,$$

and repeats with period $2\varpi\approx5.2441$. The symbol $\varpi$, a pi written another way, is the lemniscate constant.

Abel published the theory in 1827, in the second volume of Crelle’s Journal, under the title Researches on elliptic functions. The German Carl Jacobi arrived at the same ideas independently in the same year, and the two spent two years in a race of papers.

Two periods

The great surprise appears when $s$ is allowed to be a complex number. In the integral, replace $t$ by $iu$. Since $(iu)^4=u^4$ and $dt=i\,du$,

$$\int_0^{ix}\frac{dt}{\sqrt{1-t^4}}=i\int_0^x\frac{du}{\sqrt{1-u^4}}=is.$$

This means that $\operatorname{sl}(is)=i\operatorname{sl}(s)$. Since $\operatorname{sl}$ has the real period $2\varpi$, it gets an imaginary period, $2\varpi i$, for free:

$$\boxed{\operatorname{sl}(s+2\varpi)=\operatorname{sl}(s),\qquad\operatorname{sl}(s+2\varpi i)=\operatorname{sl}(s).}$$

The sine repeats in one direction only. Elliptic functions repeat in two directions of the complex plane: their values are determined by a single parallelogram, like a tile repeated over the whole floor. Abel showed that this holds for all elliptic functions, and that they obey addition formulas similar to that of the sine, $\sin(a+b)=\sin a\cos b+\sin b\cos a$.

Exercise: a pendulum swings with an amplitude of 60°. How much longer is its period than the simple formula says?

With $\theta_0=60°$, $k=\sin30°=\frac12$ and $K\left(\frac12\right)\approx1.6858$. The ratio is $\dfrac{4K}{2\pi}=\dfrac{4\cdot1.6858}{6.2832}\approx1.073$: the period is about 7% longer. A pendulum clock swinging that widely would lose almost an hour and forty minutes a day, which is why clocks use small amplitudes.

Exercise: show that the derivative of the lemniscatic sine is sl′(s) = √(1 − sl(s)⁴).

If $x=\operatorname{sl}(s)$, then $s=\int_0^x\frac{dt}{\sqrt{1-t^4}}$, and by the fundamental theorem of calculus $\frac{ds}{dx}=\frac1{\sqrt{1-x^4}}$. The derivative of the inverse is the reciprocal of the derivative: $\frac{dx}{ds}=\sqrt{1-x^4}$. It is the exact analogue of $\sin's=\cos s=\sqrt{1-\sin^2s}$, with the fourth power in place of the square.

Sources: Niels Henrik Abel, “Recherches sur les fonctions elliptiques”, Journal für die reine und angewandte Mathematik 2 (1827) and 3 (1828) · Carl Jacobi, Fundamenta nova theoriae functionum ellipticarum (1829) · Jeremy Gray, Real and the Complex: A History of Analysis in the 19th Century (Springer, 2015).

All topics on Niels Henrik Abel · ← The binomial series and the devil’s series · Dividing the lemniscate with ruler and compass →

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