{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:13:52Z","timestamp":1776842032447,"version":"3.51.2"},"reference-count":28,"publisher":"American Mathematical Society (AMS)","issue":"267","license":[{"start":{"date-parts":[[2010,1,6]],"date-time":"2010-01-06T00:00:00Z","timestamp":1262736000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Tetration\n \n \n \n F<\/mml:mi>\n F<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as the analytic solution of equations\n \n \n \n \n F<\/mml:mi>\n (<\/mml:mo>\n z<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n =<\/mml:mo>\n ln<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n F<\/mml:mi>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n F(z-1)=\\ln (F(z))<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n F<\/mml:mi>\n (<\/mml:mo>\n 0<\/mml:mn>\n )<\/mml:mo>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n F(0)=1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is considered. The representation is suggested through the integral equation for values of\n \n \n \n F<\/mml:mi>\n F<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n at the imaginary axis. Numerical analysis of this equation is described. The straightforward iteration converges within tens of cycles; with double precision arithmetics, the residual of order of 1.e-14 is achieved. The numerical solution for\n \n \n \n F<\/mml:mi>\n F<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n remains finite at the imaginary axis, approaching fixed points\n \n \n \n L<\/mml:mi>\n L<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n L<\/mml:mi>\n \n \n \u2217\n \n <\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n L^{*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of logarithm (\n \n \n \n \n L<\/mml:mi>\n =<\/mml:mo>\n ln<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n L<\/mml:mi>\n <\/mml:mrow>\n L=\\ln L<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ). Robustness of the convergence and smallness of the residual indicate the existence of unique tetration\n \n \n \n \n F<\/mml:mi>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n F(z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , that grows along the real axis and approaches\n \n \n \n L<\/mml:mi>\n L<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n along the imaginary axis, being analytic in the whole complex\n \n \n \n z<\/mml:mi>\n z<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -plane except for singularities at integer the\n \n \n \n \n z<\/mml:mi>\n ><\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n z>-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and the cut at\n \n \n \n \n z<\/mml:mi>\n ><\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n z>-2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Application of the same method for other cases of the Abel equation is discussed.\n <\/p>","DOI":"10.1090\/s0025-5718-09-02188-7","type":"journal-article","created":{"date-parts":[[2009,4,27]],"date-time":"2009-04-27T13:47:33Z","timestamp":1240840053000},"page":"1647-1670","source":"Crossref","is-referenced-by-count":13,"title":["Solution of \ud835\udc39(\ud835\udc67+1)=exp\\big(\ud835\udc39(\ud835\udc67)\\big) in complex \ud835\udc67-plane"],"prefix":"10.1090","volume":"78","author":[{"given":"Dmitrii","family":"Kouznetsov","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,6]]},"reference":[{"issue":"196","key":"1","doi-asserted-by":"publisher","first-page":"723","DOI":"10.2307\/2938713","article-title":"Infinitely differentiable generalized logarithmic and exponential functions","volume":"57","author":"Walker, Peter","year":"1991","journal-title":"Math. 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