Abstract
In this paper we will consider the tetration, defined by the equation F(z + 1) = bF(z) in the complex plane with F(0) = 1, for the case where b is complex. A previous paper determined conditions for a unique solution the case where b is real and b > e1/e. In this paper we extend these results to find conditions which determine a unique solution for complex bases. We also develop iteration methods for numerically approximating the function F(z), both for bases inside and outside the Shell-Thron region.
References
Kneser, H.: Reelle analytishe Lösungen der Gleichung φ(φ(x)) = e x und verwandter Funktionalgleichungen. J. reine angew. Math. 187, 56–67 (1950)
Lawrence, P. W., Corless, R. M., Jeffrey, D. J.: Algorithm 917: Complex double-precision evaluation of the wright omega function. ACM Trans. Math. Softw. 38, 3, 20 (2012). 1–17
Paulsen, W., Cowgill, S.: Solving F(z + 1) = b F(z) in the complex plane. Adv. Comput. Math. 43 (6), 1261–1282 (2017). https://doi.org/10.1007/s10444-017-9524-1
Kouznetsov, D.: Solution of F(z + 1) = exp(F(z)) in the complex z-plane. Math. Comput. 78(267), 1647–1670 (2009)
Trappmann, H., Kouznetsov, D.: Computation of the two regular super-exponentials to base e 1/e. Math. Comput. 81, 2207–2227 (2012). https://doi.org/10.1090/S0025-5718-2012-02590-7
Mathematics Stack Exchange, What is i exponentiated to itself i times? (2013), https://math.stackexchange.com/questions/280251
Kouznetsov, D., Trappmann, H.: Portrait of the four regular super-exponentials to base sqrt(2). Math. Comput. 79, 1727–1756 (2010). https://doi.org/10.1090/S0025-5718-10-02342-2
Doyle, A.C.: The Sign of Four. The Complete Sherlock Holmes. Garden City Publishing Company, New York (1938)
Levenson, S: Complex base tetration program. Tetration and Related Topics. Tetration Forum. http://math.eretrandre.org/tetrationforum/showthread.php?tid=729 (2012)
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Communicated by: Aihui Zhou
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Paulsen, W. Tetration for complex bases. Adv Comput Math 45, 243–267 (2019). https://doi.org/10.1007/s10444-018-9615-7
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DOI: https://doi.org/10.1007/s10444-018-9615-7