Article electronically published on January 6, 2009 SOLUTION OF F (z +1)=exp ( F (z) ) IN COMPLEX z-PLANE

Tetration F as the analytic solution of equations F (z − 1) = ln(F (z)), F (0) = 1 is considered. The representation is suggested through the integral equation for values of F 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.e14 is achieved. The numerical solution for F remains finite at the imaginary axis, approaching fixed points L, L∗ of logarithm (L = lnL). Robustness of the convergence and smallness of the residual indicate the existence of unique tetration F (z), that grows along the real axis and approaches L along the imaginary axis, being analytic in the whole complex z-plane except for singularities at integer the z <−1 and the cut at z <−2. Application of the same method for other cases of the Abel equation is discussed.