How do you find an inverse function if that inverse can only be expressed as a Taylor series expansion?

Question

For example, how do you find the inverse function of Y = X^X ? Solving for X in terms of Y can't be done using basic algebra. How do I calculate the power series expansion of the inverse function?

Alan · Answer

A Taylor series doesn't help at all for the function 
you mentioned.    
Also, for even value of  x between 0 and 1  , there are 
two y values for every x so an inverse would not be a function.  
so a Taylor series will be like the following  
So with all  different exponent levels.  the Taylor 
doesn't help find the inverse.

y = c +  (x-a)*b  + (x-a)^2*c  + (x-a)^3*d   + ...

How do you find an inverse function if that inverse can only be expressed as a Taylor series expansion?

1 Answer