Faa di Bruno's formula for inverse functions ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:08:39Z http://mathoverflow.net/feeds/question/98501 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98501/faa-di-brunos-formula-for-inverse-functions Faa di Bruno's formula for inverse functions ? Feldmann Denis 2012-05-31T15:59:19Z 2012-06-02T17:17:24Z <p>It is easy to get a expression for the nth-derivative of an inverse fuction ; starting from $(f^{-1})'=\frac{1}{f'\circ f^{-1}}$, one gets things like $(f^{-1})^{(n)}=\frac{\sum a_k\prod (f^{(n_j)}\circ f^{-1})^j}{(f'\circ f^{-1})^{2n-1}}$, with reasonably easy constraints on the $n_j$. But what are the values of the $a_k$? I believe I read somewhere this was an application of umbral calculus, but I dont see how, and inverting Faa di Bruno's formula on the identity $f\circ f^{-1}=id$ dont seem to get anywhere.</p> http://mathoverflow.net/questions/98501/faa-di-brunos-formula-for-inverse-functions/98503#98503 Answer by Abdelmalek Abdesselam for Faa di Bruno's formula for inverse functions ? Abdelmalek Abdesselam 2012-05-31T16:20:22Z 2012-05-31T16:20:22Z <p>You should be able to get a formula, first by reducing to the case where f(0)=0 and the evaluation of the derivatives (for both f and its inverse) is at 0. Then, work formally by replacing f by its Taylor-MacLaurin series at 0. The problem then becomes that of the reversion of power series. It has been done in many places and typically involves summing over trees.</p> http://mathoverflow.net/questions/98501/faa-di-brunos-formula-for-inverse-functions/98517#98517 Answer by Ira Gessel for Faa di Bruno's formula for inverse functions ? Ira Gessel 2012-05-31T20:15:09Z 2012-05-31T20:15:09Z <p>See Warren P. Johnson, Combinatorics of Higher Derivatives of Inverses, American Mathematical Monthly, Vol. 109, No. 3 (Mar., 2002), pp. 273-277, <a href="http://www.jstor.org/stable/2695356" rel="nofollow">http://www.jstor.org/stable/2695356</a></p> http://mathoverflow.net/questions/98501/faa-di-brunos-formula-for-inverse-functions/98553#98553 Answer by Liviu Nicolaescu for Faa di Bruno's formula for inverse functions ? Liviu Nicolaescu 2012-06-01T09:08:41Z 2012-06-01T09:08:41Z <p>This is sometime called the <a href="http://en.wikipedia.org/wiki/Lagrange_inversion_theorem" rel="nofollow"><em>Lagrange inversion formula</em></a>.</p> http://mathoverflow.net/questions/98501/faa-di-brunos-formula-for-inverse-functions/98664#98664 Answer by Feldmann Denis for Faa di Bruno's formula for inverse functions ? Feldmann Denis 2012-06-02T15:47:49Z 2012-06-02T15:47:49Z <p>To precise my question, I was asking for the exact values of the $a_k$. Thanks to Tom Copeland, I could find the sequence A176740 of OEIS, giving a complete answer (with useful links) to this problem.</p> http://mathoverflow.net/questions/98501/faa-di-brunos-formula-for-inverse-functions/98670#98670 Answer by Daniel Geisler for Faa di Bruno's formula for inverse functions ? Daniel Geisler 2012-06-02T17:17:24Z 2012-06-02T17:17:24Z <p>Riordan's <strong>Combinatorial identities</strong> has a chapter on partition polynomials that may be helpful. It specifically covers the question you are asking, but is in umbral calculus. </p>