The bottom side represents the real valued part of sexp*(x) where the domain is [0,3]. The range is thus [sexp*(0),sexp*(3)]. sexp*(x) is the ordinary sexp but with sexp(0) = 0.
We assume strictly rising here.
This part is estimated by a truncated real Taylor series with alternating signs.
Now the upper left side is then sexp*(ix).
the upper right side is exp^[3](sexp*(ix)) = sexp*(ix + 3).
The diagonal has lenght 3sqrt(2) and is computed by continuation of sexp*(ix) and taking sexp*((-1)^{1/4} x).
Now the red triangle has contour 0 ( because of cauchy's integral theorem ).
Therefore all sides are defined thus solvable.
Ofcourse we also require
On the bottom side :
sexp*(1) = 1 , sexp*(2) = e ,
On the diagonal side
sexp*(i+1) = exp(sexp*(i)) , sexp*(2i+2) = exp(exp(sexp*(2i))).
Now we can solve the equations and then pick one that satisfies sexp*(z+1) = exp(sexp*(z)).
regards
tommy1729
We assume strictly rising here.
This part is estimated by a truncated real Taylor series with alternating signs.
Now the upper left side is then sexp*(ix).
the upper right side is exp^[3](sexp*(ix)) = sexp*(ix + 3).
The diagonal has lenght 3sqrt(2) and is computed by continuation of sexp*(ix) and taking sexp*((-1)^{1/4} x).
Now the red triangle has contour 0 ( because of cauchy's integral theorem ).
Therefore all sides are defined thus solvable.
Ofcourse we also require
On the bottom side :
sexp*(1) = 1 , sexp*(2) = e ,
On the diagonal side
sexp*(i+1) = exp(sexp*(i)) , sexp*(2i+2) = exp(exp(sexp*(2i))).
Now we can solve the equations and then pick one that satisfies sexp*(z+1) = exp(sexp*(z)).
regards
tommy1729
