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Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function. Setting g(x) = 0 and assuming a ≠ 0 produces a quintic equation of the form:
Something more general is required for equations of higher degree, so to solve the quintic, Hermite, et al. replaced the exponential by an elliptic modular function and the integral (logarithm) by an elliptic integral. Kronecker believed that this was a special case of a still more general method. [1]
Proving that the general quintic (and higher) equations were unsolvable by radicals did not completely settle the matter, because the Abel–Ruffini theorem does not provide necessary and sufficient conditions for saying precisely which quintic (and higher) equations are unsolvable by radicals.
The general quintic may be reduced into what is known as the principal quintic form, with the quartic and cubic terms removed: + + + =. If the roots of a general quintic and a principal quintic are related by a quadratic Tschirnhaus transformation = + +, the coefficients and may be determined by using the resultant, or by means of the power sums of the roots and Newton's identities.
He subsequently spent many years on Ramanujan's formulae in the area of modular equations, mock theta functions [9] and q-series, and for some time looked after Ramanujan's lost notebook. Sometime in the late 1920s, G. N. Watson and B. M. Wilson began the task of editing Ramanujan's notebooks. The second notebook, being a revised, enlarged ...
Download as PDF; Printable version ... is a root of the equation. The full quartic can then be factorized this way: ... only ones available for quintic and higher ...