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.) |