[2sinh] using exp(x) - exp(-3/5 x)

[tommy1729]

I realize a variant of this with parabolic fixpoint (probably at 0) is the key to solving some of my older post ideas.

But not exp(x) - 1 that is not asymptotic enough.

so maybe something like

exp(x) - exp(- 3/5 x) + exp(- a x) - exp(- b x)

But that gives me the system of equations (from taylor expansion )

- a + b + 8/5 = 1
a^2 / 2 - b^2 + 8/25 > 0
- a^3 + b^3/6 + 76/375 > 0
...

b = 1 + a - 8/5
...

I am unsure.

This does not give all derivatives > 0 I think, so it kinda fails.


There are also some additional conditions.

I need to work on this.

I will come back to this later.

Sorry for being vague.

Open to suggestions !



regards

tommy1729