Fuchsian Functions Research Papers - Mathematical Analysis & Applications for Advanced Studies & Academic Research

Fuchsian functions are generalisations of elliptic functions. Elliptic functions are inverses of elliptic integrals such as the integral of 1/(1-x^2)(1-k^2x^2). This is a classical class of integrals that cannot be evaluated in terms of elementary functions. These integrals show up all the time, for instance when finding the arc-length of an ellipse (hence their name), so technical progress was prompted, such as an addition formula of Euler, but only in the 19th century was conceptual clarity brought to the matter as follows. The Riemann surface of the integrand of an elliptic integral is a torus, so the integral is ambiguous up to a linear combination of the two generating loops, w_1 and w_2, so the inverse of the integral is doubly periodic with periods w_1 and w_2. And w_1 and w_2 span a parallelogram in the complex plane so an elliptic function is periodic with respect to the generated lattice, or, as one says, automorphic with respect to its group of motions. The idea of periodicity with respect to a group had also made scattered appearances elsewhere. A particularly important case to Poincaré was second order linear differential equations. Now, if we solve such an equation by power series and take the analytic continuation around a singular point then we will in general not get back to th solution we started with; in fact, the quotient of two solutions will undergo a linear fractional transformation since any two linearly independent solutions are linear combinations of any other two. So, as in the case of elliptic functions, the fundamental object has a many-valued nature and inverting it gives a periodic function. Solving differential equations now boils down to finding functions automorphic with respect to the group of linear fractional transformations in question. These turn out to be groups of motions of tessellations of a disc by curvilinear polygons. Poincaré's wonderful insight, which came to him as he stepped onto a bus, is that these groups are in fact groups of motions in hyperbolic geometry, just as the motions associated with elliptic functions are Euclidean. So now the entire theory can be built on the foundation of hyperbolic tessellations, and the corresponding automorphic functions can the be constructed as quotients of theta functions as in the elliptic case. The sense in which these functions are "the next step" after elliptic functions is particularly clear topologically: in the elliptic case, the quotient of the plane by the group of motions of the lattice is a torus; in the Fuchsian case, the possible quotient surfaces are the orientable surfaces of higher genus. Since curves of genus 0 can be parametrised by rational functions (e.g., the circle x^2+y^2=1 by x=(1-t^2)/(1+t^2) and y=2t/(1+t^2)) and curves of genus 1 can be parametrised by elliptic functions, Poincaré was led to the "uniformisation conjecture" that higher genus curves can be parametrised by corresponding Fuchsian functions.