File:Sixth root as a composite function of cbrt and sqrt.svg

English: Diagram showing how a list monad can be used to compose complex roots of a number. sqrt and cbrt are, respectively, the complex multivalued square and cube root functions. When composed with the "bind" operator "•" they produce the sixth root function. Here z is a complex number, and the square brackets denote an array.

unit(z) := [ z ]

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(f•g)(z) := append(map(f,g(z)))

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lift(f) = f° := unit•f = f•unit

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sqrt°(z) =

= append(map(unit,sqrt(z)))

= append(map(sqrt,unit(z)))

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sxrt(z) = (cbrt°•sqrt°)(z)

= append(map(cbrt°,sqrt°(z)))