integral. Glasser's derivation This derivation due to M. L. Glasser generalizes the series method presented earlier in this article to find a solution to any trinomial equation of the form: In particular, the quintic equation can be reduced to this form by the use of Tschirnhaus transformations as shown above. Let , the general form becomes: where A formula due to Lagrange states that for any analytic function , in the neighborhood of a root of the transformed general equation in terms of , above may be expressed as an infinite series: If we let in this formula, we can come up with the root: By the use of the Gauss multiplication theorem the infinite series above may be broken up into a finite series of hypergeometric functions: and the trinomial of the form has roots A root of the equation can thus be expressed as the sum of at most N − 1 hypergeometric functions. Applying this method to the reduced Bring–Jerrard quintic, define the following functions: which are the hypergeometric functions that appear in the series formula above. The roots of the quintic are thus: This is essentially the same result as that obtained by the following method. The method of differential resolvents James Cockle and Robert Harley developed, in 1860, a method for solving the quintic by means of differential equations. They consider the roots as being functions of the coefficients, and calculate a differential resolvent based on these equations. The Bring–Jerrard quintic is expressed as a function: and a function is to be determined such that: The function must also satisfy the following four differential equations: Expanding these and combining them together yields the differential resolvent: The solution of the differential resolvent, being a fourth order ordinary differential equation, depends on four constants of integration, which should be chosen so as to satisfy the original quintic. This is a Fuchsian ordinary differential equation of hypergeometric type, whose solution turns out to be identical to the series of hypergeometric functions that arose in Glasser's derivation above. This method may also be generalized to equations of arbitrarily high degree, with differential resolvents which are partial differential equations, whose solutions involve hypergeometric functions of several variables. A general formula for differential resolvents of arbitrary univariate polynomials is given by Nahay's powersum formula. Doyle–McMullen iteration In 1989, Peter Doyle and Curt McMullen derived an iteration method that solves a quintic