Bring radical
The Bring radical of a complex number a is either any of the five roots of the above polynomial (it is thus multi-valued), or a specific root, which is usually chosen such that the Bring radical is real-valued for real a and is an analytic function in a neighborhood of the real line.
Because of the existence of four branch points, the Bring radical cannot be defined as a function that is continuous over the whole complex plane, and its domain of continuity must exclude four branch cuts.
For real argument, it is odd, monotonically decreasing, and unbounded, with asymptotic behavior
The quintic equation is rather difficult to obtain solutions for directly, with five independent coefficients in its most general form:
may be determined by using the resultant, or by means of the power sums of the roots and Newton's identities.
consisting of a quadratic and a linear equation, and either of the two sets of solutions may be used to obtain the corresponding three coefficients of the principal quintic form.
[2] It is possible to simplify the quintic still further and eliminate the quadratic term, producing the Bring–Jerrard normal form:
This method was also discovered by Jerrard in 1852,[3] but it is likely that he was unaware of Bring's previous work in this area.
[1](pp92–93) The full transformation may readily be accomplished using a computer algebra package such as Mathematica[4] or Maple.
[5] As might be expected from the complexity of these transformations, the resulting expressions can be enormous, particularly when compared to the solutions in radicals for lower degree equations, taking many megabytes of storage for a general quintic with symbolic coefficients.
A Taylor series for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows.
It may be interesting to compare with the hypergeometric functions that arise below in Glasser's derivation and the method of differential resolvents.
[citation needed] The problem is now reduced to the Bring–Jerrard form in terms of solvable polynomial equations, and using transformations involving polynomial expressions in the roots only up to the fourth degree, which means inverting the transformation may be done by finding the roots of a polynomial solvable in radicals.
This procedure gives extraneous solutions, but when the correct ones have been found by numerical means, the roots of the quintic can be written in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions (defined broadly to include Bring radicals) of a single variable — an algebraic solution of the general quintic.
Many other characterizations of the Bring radical have been developed, the first of which is in terms of "elliptic transcendents" (related to elliptic and modular functions) by Charles Hermite in 1858, and further methods later developed by other mathematicians.
In 1858, Charles Hermite[7] published the first known solution to the general quintic equation in terms of "elliptic transcendents", and at around the same time Francesco Brioschi[8] and Leopold Kronecker[9] came upon equivalent solutions.
into which any quintic equation may be reduced by means of Tschirnhaus transformations as has been shown.
may be related to the Bring–Jerrard quintic by the following function of the six roots of the modular equation (In Hermite's Sur la théorie des équations modulaires et la résolution de l'équation du cinquième degré, the first factor is incorrectly given as
The Hermite–Kronecker–Brioschi method essentially replaces the exponential by an "elliptic transcendent", and the integral
This theorem, known as Thomae's formula, was fully expressed by Hiroshi Umemura[16] in 1984, who used Siegel modular forms in place of the exponential/elliptic transcendents, and replaced the integral by a hyperelliptic integral.
This derivation due to M. Lawrence Glasser[17] 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.
, in the neighborhood of a root of the transformed general equation in terms of
James Cockle[18] and Robert Harley[19] 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 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,[20] whose solution turns out to be identical to the series of hypergeometric functions that arose in Glasser's derivation above.
[5] 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.
[23][24] In 1989, Peter Doyle and Curt McMullen derived an iteration method[25] that solves a quintic in Brioschi normal form:
Due to the way the iteration is formulated, this method seems to always find two complex conjugate roots of the quintic even when all the quintic coefficients are real and the starting guess is real.
