Trying to solve f(f(x)) = x**2  2 
Problem: Find satisfying the functional equation
Solution:
 First, it is noteworthy that the equation has simple solutions
 One can try to reduce (1) to (2) by changing "coordinates" with
 In y coordinates, eq.(1) can be rewritten as where
 Requesting eq.(2) to hold for gives or or where we denoted Now a solution g(y) of the new functional equation (9) will transform a solution (3) of eq.(2) (more precisely, the solution of the eq.(7)) into a solution of eq.(1):
 Looking for solutions of eq.(9)
 Assumption 1: is defined in the vicinity of
 Assumption 2A: and its derivatives of all orders are finite at . Then, by comparing Taylorseries expansions for the left and righthand sides of eq.9, one can conclude that all the derivatives are zeros, i.e. . The corresponding solutions, and , are of no interest for us, as these functions are not invertible.
 A singularity of at is needed.
 Assumption 2B:
 Assumption 3: at with
 A good guess provides a solution for any
 Properties:
 defined at
 has maxima at and
and one minimum
 there are two inverse functions, both defined in the interval , and taking values in the intervals and respectively
 Independently of the value, all the solutions will give the same result when fed into eq.(11), as we will see a bit later.
Therefore, we can consider the details for just one case of e.g. :

 The two inverse functions are
or for any
 Feeding the found solution into the formula (11)
According to (11) we get  the solution of eq.(1)  in three steps:
 taking : with eq.(13.1), the result is
 applying one of two functions to the result of step 1: taking e.g. , gives
 finally, applying function (12.1) for the argument resulting from the step 2:
 Remarks
 Due to specific properties of the function applied on step 3, the result does not depend on which one out of the two functions is chosen on step 2
 If the general solution (13) is taken instead of (13,1) on step 1, then an additional exponential power appears on step 1 and gets propagated on step 2. Then, on step 3, the general formula (13) is applied instead of (13.1), and this involves the extra power, which cancels the power. So the result does not change.
 The results are identical for the plus and minus signs in the formula used on step 1.
Indeed, , Therefore, changing plus to minus just exchanges the terms and on step 3.
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