Galidakis, I. N. On solving the \(p\)-th complex auxiliary equation \(f^{(p)}(z) = z\). (English) Zbl 1188.30030 Complex Variables, Theory Appl. 50, No. 13, 977-997 (2005). Summary: In this article we solve the first and second real auxiliary exponential equations using Lambert’s \(W\) function. We exhibit a class of functions which extend \(W\). We solve analytically the first, second and \(p\)-th complex auxiliary exponential equations using these functions, and give an analytic characterization of the domains of periodic points of order \(p>1\) for the complex iterated exponential \(f^{(p)}(z) = z\). We then analytically solve transcendental equations with iterated exponential terms using a similar class of functions, and finally derive exact expressions for the derivatives and integrals of all functions involved. Cited in 3 Documents MSC: 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 30D10 Representations of entire functions of one complex variable by series and integrals 30D20 Entire functions of one complex variable (general theory) 33B10 Exponential and trigonometric functions 30B10 Power series (including lacunary series) in one complex variable 30-04 Software, source code, etc. for problems pertaining to functions of a complex variable 33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) Keywords:Auxiliary equation; Lambert’s W function; Infinite exponential; Transcendental equation; Entire function; Fixed point; Periodic point PDFBibTeX XMLCite \textit{I. N. Galidakis}, Complex Variables, Theory Appl. 50, No. 13, 977--997 (2005; Zbl 1188.30030) Full Text: DOI