These Infinite Power Towers Have a Surprise Ending

Do infinite power towers actually converge?

In this video, we compare two fascinating expressions:
i^(i^(i^(…)))
and
(−i)^(−i^(−i^(…)))

At first glance, both towers look chaotic. But using complex analysis, Euler’s formula, and the Lambert W function, we uncover their hidden structure and determine whether they converge — and if so, where.

You’ll see:
• How infinite exponentiation becomes a fixed-point problem
• Why logarithms turn iteration into geometry
• How the Lambert W function solves equations of the form
  u · e^u = k
• Why both towers converge to complex conjugates
• A numerical visual verification showing convergence step by step
This video combines rigorous mathematics with clean visual intuition, making advanced ideas in complex analysis both accessible and visually compelling.

🧠 Topics Covered
Infinite power towers
Complex exponentiation
Fixed points and convergence
Euler’s formula
Lambert W function
Complex conjugates
Iterative maps in the complex plane

🔍 Why This Is Interesting
Infinite exponentiation behaves very differently in the complex plane.
Even a small change — replacing i with −i — leads to a deep and surprising result.


🏷️ Tags / Keywords
infinite power tower, complex exponentiation, i power tower, minus i power tower, lambert w function, complex analysis, convergence, fixed point iteration, euler formula, complex conjugates, math visualization, manim animation, advanced mathematics