The Birth of Elliptic Integrals & Elliptic Functions: A Brief Historical Overview

In this post, we embark on a concise yet illuminating journey through the historical and mathematical evolution of elliptic integrals and elliptic functions—concepts that lie at the heart of modern analysis, number theory, and algebraic geometry.

Historical Origins: From Curves to Complexities

The origin of elliptic integrals traces back to one of the oldest problems in mathematics: computing arc lengths of curves. In particular, attempts to compute the arc length of an ellipse led to integrals that could not be expressed using elementary functions—thus earning the name elliptic integrals. Yet, ellipses are not the only curves that give rise to such integrals. Arc length of the lemniscate, defined by the polar equation $r^2=\cos(2\theta)$ , also leads to integrals of a similar nature.

These integrals, involving square roots of polynomials of degree 3 or 4, came to be recognized as a distinct class—now known as elliptic integrals.

Form of Elliptic Integrals

Elliptic integrals are typically expressed in the general form:

$$\int R(t,\sqrt{P(t)})dt$;\space($R:$\space Rational Function,\space$P:$\space Cubic/Quartic Polynomial with no repeated roots.)$

One interesting fact about these elliptic integrals is that the integrals are not expressible using elementary functions. A classical example is: $\large$E(a):=\int\limits_0^a\frac{dt}{\sqrt{1-t^4}}$$ which arises in the context of the lemniscate. This integral cannot be evaluated using elementary functions, making it transcendental. Those who are interested in taking a look at the detailed calculation can find it here.

From Integrals to Functions: A Shift in Perspective

Mathematicians wanted to find addition theorems for elliptic functions. This was initially their main focus of investigation. More precisely, they wanted to answer questions like the following:
$Given\space$a,b$\space for which\space$c,$\space does\space$E(a)+E(b)=E(c)$\space hold?$
However, the early pioneers did not view these elliptic integrals as functions. Initial investigators such as Euler studied these integrals primarily in their definite form. Rather they wanted to solve for $c$ such that: $\int_0^a\frac{dt}{\sqrt{1-t^4}}+\int_0^b\frac{dt}{\sqrt{1-t^4}}=\int_0^c\frac{dt}{\sqrt{1-t^4}}$ ; for given value of $a,b$ .
In fact, both Euler and Fagnano worked of the problem called Doubling the Lemniscate. Given that $\int_0^a\frac{dr}{\sqrt{1-r^4}}=2\int_0^b\frac{dr}{\sqrt{1-r^4}}$ , the doubling problem investigates the relation between the parameter values $a,b$ . It is evident that, the doubling problem is a special case of the more general addition theorem. An interesting result in regards to addition theorems of elliptic functions is the one due to Euler:

$Let\space$E$\space be an elliptic integral and let\space$E(a)+E(b)=E(c)$,\space then\space$c$\space depends only algebraically on\space$a$\space and\space$b$.$

Later on Mathematicians such as Abel, Jacobi and Legendre contributed to the study of elliptic integrals. Legendre was the first to view elliptic integrals as functions, he regarded integrals like $E(a):=\int_0^a\frac{dt}{\sqrt{1-t^4}}$ as elliptic functions. However, the modern terminology varies from Legendre’s terminology.

One of the central question in this period was the inversion problem which asks: Given a value of the integral, what is the corresponding value of the upper limit of integration? This inversion problem gradually became the main object of study. Rather than focusing solely on the integral $y=\int_0^x\frac{1}{\sqrt{1-t^4}}dt$ mathematicians began to ask: Can we define a function $x=f(y)$ that inverts this process?
This led to the concept of elliptic functions—functions that arise as inverses of elliptic integrals.

In modern days, we use the term elliptic functions to denote the inverses of elliptic integrals, for example:
$E=E(a)=\int_0^a\frac{dt}{\sqrt{1-t^4}}\implies a=\mathcal C(E)$
is called an inversion of the elliptic integral $E$ and the function $a=\mathcal C(E)$ is called its inversion.

The Contributions of Legendre, Abel, and Jacobi

Niels Henrik Abel and Carl Gustav Jacobi revolutionized the subject by studying the inverses of elliptic integrals. They discovered inversion theorems of elliptic functions independently. For example, the inversion of the integral $E(a):=\int_0^a\frac{dt}{\sqrt{1-t^2}}$ will be $a=\sin(E)$ .

Adrien-Marie Legendre was the first to systematically classify elliptic integrals into three types.

$$F(\phi,k)=\int_0^\phi\frac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}}$\space(First kind)$

$$E(\phi,k)=\int_0^\phi\sqrt{1-k^2\sin^2(\theta)}d\theta$\space(Second kind)$

$$\Pi(n;\phi,k)=\int_0^\phi\frac{d\theta}{(1-n\sin^2\theta)\sqrt{1-k^2\sin^2\theta}}$\space(Third kind)$

Legendre also tabulated extensive numerical of values for elliptic integrals, which were essential before the era of computational tools. Jacobi introduced elliptic functions such as sn, cn, and dn, which are now standard in the theory.

We can say that addition theorems of elliptic integrals are generalizations of the addition formulas of trigonometric functions. Also, addition theorem allows us to extend the domains of these integrals to the complex numbers $\mathbb C$ . When we extend the domains of these elliptic integrals to the complex numbers, the function become doubly periodic i.e., they repeat in two directions in the complex plane. This double periodicity means they are naturally defined as functions on the torus.

In fact, Jacobi and Liouville investigated deeply to find and classify all doubly-periodic elliptic functions.

The Role of Complex Analysis and contributions of Riemann

At the time of Legendre and Jacobi, complex analysis was still in its infancy. As mathematicians tried to extend elliptic integrals to complex arguments, they encountered serious difficulties: the integrals became path-dependent due to the presence of branch points, leading to multi-valued functions.. For example, if we extend $E(a):=\int_0^a\frac{dt}{\sqrt{1-t^4}}$ to complex numbers, then immediately one can see that the integral becomes path-dependent as there are several path to go from $0$ to $a$ , and thus there is a serious problem with well-definedness of the function.

The breakthrough came with Bernhard Riemann, who introduced the concept of Riemann surfaces to resolve the ambiguity. To deal with multivalued functions $\mathbb C\to\mathbb C$ defined on the complex plane like $z\mapsto\sqrt{z}$ , Riemann considered them as single-valued functions on a more sophisticated domain—a branched cover of the complex plane known as a Riemann surface and considered $z\mapsto\sqrt{z}$ to be defined from $S\to\mathbb C$ , where $S$ is the Riemann surface corresponding to $\sqrt{z}$ .

Riemann Surface of the complex square root function

This approach transformed the study of complex integrals and allowed a rigorous foundation for the theory of elliptic functions.

Beyond Elliptic: The Birth of Abelian Functions

Riemann went further and generalized elliptic functions to Abelian functions, which are inverses of Abelian integrals, which have the general form:

$$\int R(t,\omega)dt$\space with\space$F(t,\omega)=0$\space($R$:\space Rational Function,\space$F$:\space inrreducible polynomial)$
Thus, elliptic functions turn out to be a special case of abelian functions where $\omega=-\sqrt{P(t)}$ .
The modern way to define abelian integral is through the integral of a differential 1-form $\omega$ on an algebraic curve i.e. $\int_P_0^P\omega$ . These integrals are inherently path-dependent, and their multivalued nature is captured by periods $\int_P_0^P\omega$ —complex numbers arising from integrating around loops on the Riemann surface. The collection of such periods forms a lattice, and the image of the integral lives in a complex torus known as the Jacobian variety:

$\large$\int:H_1(C,\mathbb Z)\to\mathbb C^g/\Lambda$$

$where\space$C$\space is the Riemann surface of genus\space$g$,\space and\space$\Lambda$\space is the period lattice.$

Elliptic functions, in this framework, correspond to the special case where the underlying Riemann surface has genus 1.

Closing Thoughts

The theory of elliptic integrals and functions beautifully intertwines classical analysis, algebra, geometry, and topology. From the arc length of an ellipse to the algebraic structures of tori and Jacobians, this journey is a testament to the power of abstraction and the unity of mathematics.

As Riemann envisioned, to understand functions, one must sometimes first understand spaces.

Manifolds Unfolded