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picturing method.
#2
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
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Messages In This Thread
picturing method. - by tommy1729 - 07/02/2014, 08:30 AM
RE: picturing method. - by tommy1729 - 07/02/2014, 12:27 PM
RE: picturing method. - by tommy1729 - 07/02/2014, 08:14 PM
RE: picturing method. - by tommy1729 - 07/02/2014, 09:00 PM
RE: picturing method. - by tommy1729 - 07/03/2014, 08:23 AM

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