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On solving the \(p\)-th complex auxiliary equation \(f^{(p)}(z) = z\). (English) Zbl 1188.30030

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.

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.)
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