most recent 30 from http://mathoverflow.net 2013-05-22T12:52:04Z http://mathoverflow.net/feeds/user/20825 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86450/taylor-series-of-bring-radical Ziphil 2012-01-23T14:21:38Z 2012-02-15T23:11:46Z

<p>I learned a root of quintic equation $ x ^5 + x + a = 0 $ can be expressed as Taylor series.</p> <p>According to <a href="http://en.wikipedia.org/wiki/Bring_radical#Series_representation" rel="nofollow">this page</a>, the inverse function of $ f (x) = x ^5 + x $ is expressed as $$ f ^{- 1} (x) = \sum ^\infty _{k = 0} \begin{pmatrix} 5 k \\ k \end{pmatrix} \frac{(- 1) ^k x ^{4 k + 1}}{4 k + 1}, $$ but I can't understand why this expression is correct.</p> <p>I think <a href="http://en.wikipedia.org/wiki/Lagrange_inversion_theorem" rel="nofollow">Lagrange Inversion Theorem</a> and <a href="http://en.wikipedia.org/wiki/General_Leibniz_rule" rel="nofollow">General Leibniz rule</a> may be the clue.</p> <p>Please tell me why the above formula is correct.</p>