Searching for an asymptotic to exp[0.5]

[tommy1729]

I told mick to consider fake( f ) = fake( f g )/ g.

Or perhaps even fake ( f ) = fake ( f g ) / fake ( g ).

That seems like a powerfull idea at first sight.

So fake ( ln(x^2+1) ) = fake ( ln (x^2+1) exp(x^2) ) exp(-x^2).

Naturally I wonder now about

fake ( exp^[0.5](x) ) vs fake ( exp^[0.5](x) exp(x) ) exp(-x).

I assume we cannot keep the property of positive derivatives for
fake ( exp^[0.5](x) exp(x) ) exp(-x) , but still it seems intresting.

Also crossing my mind : d/dx fake ( f ) = fake ( d/dx f ).

And then there are least squares ideas.


A theoretical question : Lets write fake+ for an asymptotic with positive derivatives,

then does there Always exist a g(x) such that for any entire f(x) we have

fake+ ( f ) = fake+ ( f g ) / g

?

regards

tommy1729