Quintic equation

In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. For example:

Bring radicals

As a function of the complex variable t, the roots x of

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:x^5 - 5x - 4t = 0,

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have branch points where the discriminant 800000(t4 - 1) is zero, which means at 1, -1, i and -i. Monodromy around any of the branch points exchanges two of the roots, leaving the rest fixed. For real values of t greater than or equal to -1, the largest real root is a function of t increasing monotonically from 1; we may call this function

Related Topics:
''i'' - Monodromy

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the Bring radical, BR(t). By taking a branch cut along the real axis from minus infinity to -1, we may extend the Bring radical to the entire complex plane, setting the value along the branch cut to be that obtained by analytically continuing around the upper half-plane.

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More explicitly, let a_0 = 3, a_1 = {1over100}, a_2 = -{27over400000}, a_3 = {549/800000000}, with subsequent ai defined by the recurrence relationship

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:a_{n+4} = -{rac {185193}{5278000}},{rac {2,n+5}{n+4}}a_{n+3}

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:-{rac {9747}{

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52780000}},{rac {10,{n}^{2}+40,n+39}{ left( n+4 ight) left(

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n+3 ight) }}a_{n+2}

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:-{rac {57}{52780000}},{rac { left( 2,n+3

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ight) left( 10,{n}^{2}+30,n+17 ight) }{ left( n+4 ight)

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left( n+3 ight) left( n+2 ight) }}a_{n+1}

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-{rac {1}{6597500000}},{rac { left( 5,n+11 ight) left( 5,n+7 ight) left( 5,n+3

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ight) left( 5,n-1 ight) }{ left( n+4 ight) left( n+3

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ight) left( n+2 ight) left( n+1 ight) }}a_n.

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For complex values of t such that |t - 57| 

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:operatorname{BR}(t) = sum_{n=0}^infty a_n (t-57)^n,,

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which then can be analytically continued in the manner described.

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The roots of x5 - 5x - 4t = 0 can now be expressed in terms of the Bring radical as

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:r_n = i^{-n} operatorname{BR}(i^n t)

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for n from 0 through 3, and

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:r_4 = -r_0-r_1-r_2-r_3

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for the fifth root.

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