Carl Gustav Jacob Jacobi: Fundaments of a new theory of elliptic functions (1829)

(1804-1851)

The development of mathematics has often been one of generalisation: concrete problems have demanded development of very abstract tools that can then be applied to various other fields. One clear example is integration. Originally developed as a tool for calculating lengths, areas and volumes, it could then be used in various other contexts where calculating limits of infinite sums with high precision was required.

Another aspect of the development of mathematics, then, has been that these very abstract tools themselves provide new problems and topics for discussion. For instance, going back to the example of integration, if you progress beyond Calculus 101, you soon learn that not all integrals can be solved through those neat formulas given in the textbook, but in the worst case scenario you have to go to the definition of integral and approximate it through various finite sums. Of course, mathematicians have found various new tools for simplifying this numerical process in individual cases. A particularly interesting case concerns the so-called elliptic integrals.

The very concept of an elliptic integral belies an origin in quite concrete geometric problems: measuring the length of pieces of a curve called ellipsis (picture an elongated version of a circle). In truth, this example is just one version of elliptic integrals, the unifying elements being certain simplicity in the formal characteristics of the base function integrated (to put it very briefly, they involve nothing more complex than fractions with denominator a square root of polynomial of third degree). These kinds of integrals are already harder than those in elementary textbooks and their exact values can often be just approximated. The question is how to simplify this process of approximation.

First step in this simplification was provided by Adrien-Marie Legendre, who showed that all the various elliptic integrals could be reduced to three paradigmatic cases - already a huge improvement. Another important step was to note that these elliptic functions could be expressed in terms of two parameters: an angle called an amplitude of the integral and a number called module. In the particular case of ellipsis, the amplitude describes the angle the x-axis makes with a line joining the origin and the tip of its particular arc, while the module describes how elongated the ellipsis is (0 being the case of a proper circle).

Jacobi’s Fundamenta nova theoriae functionum ellipticarum took the simplification a few steps further. The basic problem Jacobi set out to solve was to show that an elliptic integral with a seemingly more complex structure could be reduced to an elliptic integral of a less complex kind, provided some relation of them, expressible through relatively simple algebraic means, was shown to hold between them. In other words, by knowing that such a relation existed between the two integrals, one could calculate the value of the more complex on the basis of the simpler one. Jacobi calls this transformation of the elliptic integral.

Now, the question was how to determine this relatively simple transformation. Jacobi showed that this question was essentially the same as determining a certain type of relation between the modules of the two integrals (what he called modular equation). In principle, this modular equation could be calculated through algebraic means, but in practice, the more complex the equation changes, the more cumbersome this calculation becomes. Jacobi’s solution is to go a bit further in the level of abstraction and to construct a more general rule picking a series of suitable modules that can be linked with such transformations.

Jacobi’s derivation of this rule is based on the second parameter, the amplitude, and the so-called elliptic functions, which can be defined on the basis of the amplitudes. To give a rough idea of these elliptic functions, we can compare them with simpler trigonometric functions. It is a well-known fact that trigonometric functions can be described in terms of a unit circle and angles set up on its centre. Elliptic functions can also be described in terms of an angle set up on a centre of an ellipse.

It was just inevitable that this new tool - elliptic functions - became a topic interesting in itself. Thus, the second half of Jacobi’s work is dedicated to the study of elliptic functions, which, just like trigonometric functions, are a source of many beautiful equations. A particular question Jacobi dealt with was how to express these functions as infinite series. In effect, this was yet again a way to find more and more good approximations for elliptic functions. Finding these infinite series required the introduction of yet another tool: the so-called theta functions, which are a certain type of series of complex numbers - and the development of the mathematics continued.