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→Extension to real numbers: When this is defined for 0<''x''<1 the whole function easily follows for all ''x''>-2 |
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Extending x^^b to real numbers b is straightforward and gives, for each natural number b, a '''super-power function''' f(x) = x^^b.
*it is correct for natural numbers x
*a^^(b+1) = 2^(a^^b)
*it is a smooth function, monotonically increasing
When this is defined for 0<''x''<1 the whole function easily follows for all ''x''>-2
See http://home.earthlink.net/~mrob/pub/math/ln-notes1.html#real-hyper4 for attempts to extend tetration to real numbers.
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