page_title: Tetration
section_title: Linear approximation for real heights
hierarchical_section_title: Tetration / Extensions / Extensions of the domain for different heights / Real heights / Linear approximation for real heights
caption_attribution_description: English: Author: Andrew Robbins. I made this with Mathematica. I grant fair use to this image. This is an image of real tetration using the linear approximation.
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: A linear approximation (solution to the continuity requirement, approximation to the differentiability requirement) is given by:
  hence:
  and so on. However, it is only piecewise differentiable; at integer values of x the derivative is multiplied by . It is continuously differentiable for  if and only if . For example, using these methods  and 
  A main theorem in Hooshmand's paper states: Let . If  is continuous and satisfies the conditions:
   is differentiable on (−1, 0),
   is a nondecreasing or nonincreasing function on (−1, 0),
  then  is uniquely determined through the equation
  where  denotes the fractional part of x and  is the -iterated function of the function .
  The proof is that the second through fourth conditions trivially imply that f is a linear function on [−1, 0].
  The linear approximation to natural tetration function  is continuously differentiable, but its second derivative does not exist at integer values of its argument. Hooshmand derived another uniqueness theorem for it which states:
  If  is a continuous function that satisfies:
   is convex on (−1, 0),
  then . [Here  is Hooshmand's name for the linear approximation to the natural tetration function.]
  The proof is much the same as before; the recursion equation ensures that  and then the convexity condition implies that  is linear on (−1, 0).
  Therefore, the linear approximation to natural tetration is the only solution of the equation  and  which is convex on (−1, +∞). All other sufficiently-differentiable solutions must have an inflection point on the interval (−1, 0).
caption: x
            
          
          e
        
      
      {\displaystyle \,{}^{x}e}
    
   using linear approximation.
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