We know that we can express the Abel function of at by

where is the Schröder function of at , where .

This means that the inverse of the Abel function which is actually the superexponential can be expressed by

with appropriate translation along the x-axis choosen such that . Here

The coefficients of , , , can be recursively computed from the equation

(*) ,

by the composition formula

where .

Now we put formula (*) in:

On the right side only occurs for in , namely in the summand . Thatswhy we have the recursive formula:

So each coefficient of is a polynomial in ( base) and a rational function in ( fixed point).

and upto translation along the x-Axis we have .

where is the Schröder function of at , where .

This means that the inverse of the Abel function which is actually the superexponential can be expressed by

with appropriate translation along the x-axis choosen such that . Here

The coefficients of , , , can be recursively computed from the equation

(*) ,

by the composition formula

where .

Now we put formula (*) in:

On the right side only occurs for in , namely in the summand . Thatswhy we have the recursive formula:

So each coefficient of is a polynomial in ( base) and a rational function in ( fixed point).

and upto translation along the x-Axis we have .