Exploring the Infinite Tetration of Imaginary Numbers: Convergence and Trajectories

In the vast landscape of complex analysis, one fascinating topic involves the infinite tetration of imaginary numbers. While there isn't a closed-form formula for the #x78;n#x2061;(i), we can explore the concept using infinite tetration. This article delves into the convergence of these operations and the trajectories they create, providing a comprehensive guide for understanding this unique mathematical concept.

Convergence of Infinite Tetration

Let x represent the infinite tetration of i. If x converges, it satisfies the equation x ix. This convergence point x is a fixpoint of the tetration of i. To solve this, we use the complex Lambert W function W, the inverse of the function y xe^x, where x Wy.

General Case Solution

We start by solving the generic case:

Given the equation x i^x 1 This implies xe^xln a 1 Consequently, xln ae^xln a ln a Which simplifies to xln a Wln a Finally, x frac{Wln a}{ln a}

Substituting a frac{1}{i}, we use the principal logarithm:

Thus, x frac{Wleft(-frac{pi i}{2}right)}{-frac{pi i}{2}} Which simplifies to x approx 0.43828293 - 0.36059247i

Iterative Process and Trajectories

To see how the infinite tetration evolves, we iterate the process starting from x_0 i and use the relation:

x_n ix_{n-1} This transforms to x_n e^{x_{n-1}frac{pi i}{2}}

The trajectory of these iterations on the complex plane shows a spiral movement around the point x_{infty}.

Real and Imaginary Parts

Separating the real and imaginary parts, let a_k Real x_k and b_k Imag x_k, so:

a_0 b_0 0 1 a_n b_n lefte^{-b_{n-1}frac{pi}{2}}coslefta_{n-1}frac{pi}{2}right} e^{-b_{n-1}frac{pi}{2}}sinleftb_{n-1}frac{pi}{2}right}

The maximum distance from x_n to the origin is determined by e^{- b_{n-1}frac{pi}{2}}, which depends solely on the imaginary part of the previous point. For example, the first point i is transformed into a distance of e^{-frac{pi}{2}} approx 0.2079.

Angles and Trajectories

The angles between successive points vary but converge to about 50.55477 degrees, or 0.2808598 pi. The first five angles are about 78.288, 40.869, 42.058, 68.073, and 39.757 degrees.

Conclusion

The trajectory of the iterated tetration of i on the complex plane is a fascinating exploration of mathematics, showcasing the complex dynamics of infinite tetration and the patterns that emerge. This study not only deepens our understanding of complex analysis but also provides a visual and numerical insight into the convergence and behavior of these operations.