|San José State University|
& Tornado Alley
of a Complex Number
An infinite exponentiation is something which is raised to a power which is something raised to a power ad infinitum> Suppose
However a little manipulation turns it into a seemingly trivial problem. The manipulation is to take the G-th root of both sides giving
Now if we want a value of a that gives G as a solution we need only take the G-th root of G and we have the answer. For example, for G=2, a=2½=√2. Thus
A previous study worked out the analysis when G is a real number It was found that there is convergence if and only if a<e1/e where e=2.721828….
When a and G are complex number the equation a=G1/G is actually two equations. Let G=R·eiΦ and a=r·iθ. Then
The RHS may be rearranged in terms real and imaginary components as
The real component is maximized where d(ln(r))/dR = 0; i.e.,
(To be continued.)
Setting this equal to zero
The maximum value of the function G1/G is then e1/e.
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