Searching for an asymptotic to exp[0.5]

[tommy1729]

Dear Sheldon

I read your answer @stackexchange.

I assume your sequences are the derivatives and not the coefficients ?
Otherwise it seems divergent.

I intended to tell mick about the integral of (1-exp(x))/x to get a fake ln(x) or fake 1/x.

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And then I started wondering about x fake(1/x) , exp(fake(ln(x)) - x etc.
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I mentioned (1-exp(x))/x before.
However apparantly that was already given a long time at mathoverflow.

(as your link proves).

Its easy to get a fake sqrt from a fake ln.

So the main questions are these :

1) which is the best fake ln(x) ?
integral (1-exp(-x)/x) - C or C_2 * +fake(ln(x) exp(x)) exp(-x) ?

( I use +fake to refer to "our" method of fake functions ; with the positive derivatives )

2) Is there an analogue for the integral of (1 - exp(-x))/x to create a fake sqrt(x) ?
( I do NOT mean exp(fake(ln(x)/2) )

Answers to these will lead to new questions.

I want to remark that +fake(sqrt(x) exp(x)) exp(-x) is very likely a better approximation then exp(+fake(ln(x) exp(x)) exp(-x) / 2).

I also notice we can add variables to our ideas :

1) which is the best fake ln(x) ?
integral(1-exp(- A x)/x) - C or C_2 *+fake(ln(x) exp(B x)) exp(-B x) ?

Which leads to more intresting questions.

Anyways the most intresting is imho :

2) Is there an analogue for the integral of (1 - exp(-x))/x to create a fake sqrt(x) ?


regards

tommy1729