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(5) A( tanh(A(x)) ) = atanarctan(x).
(6) A( tan(A(x)) ) = atanharctanh(x).
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Note that A( A( tan( tanh(x) ) ) ) is NOT equal to x; the composition of these functions is not comutativecommutative.
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The series reversion of the e.g.f. is defined by A280793.
The e.g.f. as a series with reduced fractional coefficients begins:
A(x) = x + 1/30*x^5 + 5/4536*x^9 + 479/555984*x^13 - 883111/855017856*x^17 + 1014203909/361219896576*x^21 - 5103375762413/435183970636800*x^25 + 77553540368447155/1092875131729446912*x^29 +...
Series_Reversion( A(x) ) = x - 4*x^5/5! + 1616*x^9/9! - 10233664*x^13/13! + 605781862656*x^17/17! - 195074044306023424*x^21/21! + 226963189334487889924096*x^25/25! - 745095268828143694162593398784*x^29/29! + 5876637899238904537105181354518183936...+ A280793(n)*x^33(4*n-3)/33(4*n-3)! +...
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The series reversion of A(x) = tan(A(tanh(x))) = tanh(A(tan(x))), and begins:
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