Can Fuchsian functions solve the general equation of degree n? In the classic textbook Introduction to the Theory of Equations (Conkwright, 1941), on p. 85, the author writes that “the algebraic solution of the general equation of degree n is impossible if n is greater than four. By this we mean that it is not possible to find the exact values of the roots of every equation of degree n (n > 4) by performing upon the coefficients a finite number of additions, subtractions, multiplications, divisions, and root extractions.”
That proposition is considered proven by the Abel-Ruffini theorem of 1824, as Conkwright surely knew. Tantalizingly, though, he goes on to say – without elaborating – that “the general equation of degree n has been solved in terms of Fuchsian functions.” And there, it seems, the trail ends. A web search of about an hour has yielded nothing more than various restatements of the problem.
Two questions:
(1) Can anyone state, in a form suitable for reduction to a computer algorithm, a solution or family of solutions of the general equation of degree n (whether based on Fuchsian functions or not)?
(2) Would your solution(s) yield theoretically exact values, or only converging approximations?
This is my first post to your site, and I apologize in advance if I've overlooked answers right under my nose. These questions have, however, stumped my math department chair. (I’m on loan to him – I normally teach humanities, but I have an engineering background and I was asked to fill in for some math courses.)
 A: I'm not familiar with Fuchsian functions, but at the end of Mumford's "Tata Lectures on Theta II" there's an article by Hiroshi Umemura which explains how to arrive at exact solutions using theta functions.  Part of it is available on Google Books here.
A: Fuchsian functions are described on wikipedia

(1) Can anyone state, in a form
  suitable for reduction to a computer
  algorithm, a solution or family of
  solutions of the general equation of
  degree n (whether based on Fuchsian
  functions or not)?

I had prepared notes on this topic sometime ago so here goes!  There are three principal ways to solve an algebraic equation of degree n:


*

*Algebraically by using radicals
    in a number field.  This method
    doesn't work if the Galois group of
    the equation is not solvable, which
    happens for general equations beyond
    degree 4.   In terms of differential
    equations, the method is analogous to
    integrating by quadrature if the
    Automorphism group(Lie group) is
    solvable.  For more on this, see Cox's book on Galois theory and Gaal's book on Galois theory Sec 4.5

*Transcendental - by reducing
equation to some familiar "modular
equation" : The genus of the function which is used as the solution generally depends on the degree of the equation.  (See Hilbert's 13th problem) For degree {2, 3, 4},
you use trigonometric functions.  For
degree 5, (use genus 1) elliptic
functions.  For degree 6, you need
genus 2 theta functions.  For degree
7, depending on the Galois group,
requires hyperelliptic or theta
functions of genus 3.  Beyond degree
7, you need to see the Umemura paper
mentioned by Charles Seigel in his
answer.  I have omitted some information here but for general computational references, see Bruce King's book Beyond Quartic. and McKean Moll's book Elliptic Curves Chapter 5 

*Solution by complex dynamics : You
have to find an iterative map working in a function
field such that attractor(map)=root(equation).   The Newton-Raphson method
which we apply for degree 2 and 3
equations is actually a primitive
version of this method.  You can see the Julia set for Newton's method on wolfram.  For more of this
method, see the papers of Scott
Crass and Doyle McMullen.  A general reference for the method of complex dynamics is Shurman's book Geometry of the quintic.  He illustrates the similarities between this method and the method of adjoining radicals in a number field.  In both cases, you are building an extension over a number/function field.  The number field criterion says that says that splitting field can be constructed if Galois group is solvable while the function field criterion says the that Galois group must be nearly solvable (i.e. nearly solvable implies factor groups must be Mobius groups - rotation groups of the sphere)

A: you can see this paper:on fuchsian difffrential equations reducible to hypergeometric equationsby linear transformations.By Tosihusa kimura.
http://fe.math.kobe-u.ac.jp/FE/Free/vol13/fe13-17.pdf
