Essay: Memoir on Fuchsian Functions

Context and Goals
Poincaré's 1881 Memoir on Fuchsian Functions established a new class of complex functions now known as automorphic or Fuchsian functions, connecting problems from complex analysis, ordinary differential equations, and group theory. The work grew out of studies of linear differential equations with regular singular points, earlier studied by Lazarus Fuchs, and from Poincaré's exploration of discrete groups of linear fractional transformations. A central aim was to construct and classify nontrivial functions invariant under such groups and to understand their analytic and geometric structure.
The memoir set out to show how discrete groups of Möbius transformations acting on the upper half-plane or unit disk produce rich families of single-valued analytic functions, to describe methods of building these functions explicitly, and to reveal the deep relations between monodromy of differential equations and geometric tilings of the hyperbolic plane.

Main Contributions
A major achievement was the systematic construction of Fuchsian functions via what came to be called Poincaré series: summation over a discrete group of suitably chosen seed functions to produce automorphic functions that are invariant under the group's action. Poincaré investigated convergence and analytic properties of these series, giving existence proofs for nonconstant automorphic functions attached to many discontinuous groups of linear fractional transformations.
The memoir also introduced the notion of a fundamental domain and the idea of tessellating the hyperbolic plane by images of that domain under the group. These geometric pictures made explicit how the analytic behavior of automorphic functions is governed by the combinatorics and geometry of the underlying group, tying together function theory and non-Euclidean geometry.

Techniques and Ideas
Poincaré combined classical complex analysis with group-theoretic methods and differential-equation considerations. He exploited the relation between second-order linear differential equations with regular singular points and monodromy representations to connect analytic continuation and group actions: the monodromy group of a differential equation can be realized by linear fractional transformations that act discontinuously on the upper half-plane. This perspective clarified how automorphic functions arise naturally as single-valued functions on quotient Riemann surfaces.
Analytic tools such as series constructions, convergence estimates, and the study of singularities were married to geometric constructions like fundamental polygons and side-pairing transformations. Poincaré emphasized the use of hyperbolic geometry to control geometric and analytic phenomena, anticipating later formalizations of the Poincaré metric and the deep link between curvature and function theory.

Examples and Geometry
Concrete examples in the memoir include groups closely related to the modular group and functions analogous to classical elliptic modular functions. Poincaré demonstrated how different choices of generating transformations and fundamental polygons produce distinct function fields and Riemann surfaces, and he analyzed the location and nature of singularities and branching behavior through the group action.
The geometric language of polygons and tilings provided intuition about cusp points, elliptic fixed points, and the compactification of quotient surfaces. Visualizing the upper half-plane tessellated by images of a fundamental polygon made transparent relationships between group generators, side identifications, and the topology of the resulting quotient surface.

Impact and Legacy
The Memoir on Fuchsian Functions launched a program that evolved into the modern theory of automorphic forms, Fuchsian and Kleinian groups, and aspects of the uniformization of Riemann surfaces. Methods introduced there, Poincaré series, fundamental domains, and the interplay between differential equations and monodromy, became foundational tools across complex analysis, algebraic geometry, and number theory.
The work seeded later developments such as the uniformization theorem, the theory of modular forms, and the modern study of discrete subgroups of Lie groups. Beyond specific theorems, the memoir reshaped mathematical practice by showing how algebraic, analytic, and geometric viewpoints can be fused to produce powerful new theories.