page_title: Tetration
section_title: Complex bases
hierarchical_section_title: Tetration / Extensions / Extension of domain for bases / Complex bases
caption_attribution_description: English: Tetration by period. This is a fractal of tetration showing the points in the complex plane that are periodic under iteration. The red kidney shaped area has a period of one and is the area of convergence of the power tower function. The small yellow circle in the depression to the left of the red area is period two and contain zero. The large green area is period three. This fractal is similar to the Mandelbrot fractal except it is an exponential map instead of a quadratic map. The fractal was created using Fractint and shows from -4.5 to 3.5 of the real axis and from –3i to 3i on the imaginary axis.
context_page_description: In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- and iteration.
  Under the definition as repeated exponentiation, the notation means, where n copies of a are iterated via exponentiation, right-to-left, I.e. the application of exponentiation times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".
  Tetration is also defined recursively as
  ,
  allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.
  The two inverses of tetration are called the super-root and the super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.
  Tetration is used for the notation of very large numbers.
context_section_description: Since complex numbers can be raised to powers, tetration can be applied to bases of the form z = a + bi (where a and b are real). For example, in ⁿz with z = i, tetration is achieved by using the principal branch of the natural logarithm; using Euler's formula we get the relation:
  This suggests a recursive definition for ⁿ⁺¹i = a′ + b′i given any ⁿi = a + bi:
  The following approximate values can be derived:
  Solving the inverse relation, as in the previous section, yields the expected ⁰i = 1 and ⁻¹i = 0, with negative values of n giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit 0.4383 + 0.3606i, which could be interpreted as the value where n is infinite.
  Such tetration sequences have been studied since the time of Euler, but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the infinitely iterated exponential function. Current research has greatly benefited by the advent of powerful computers with fractal and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.
caption: Tetration by period
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