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 Taylor-series expansions for the left- and right-hand 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:
  1. taking : with eq.(13.1), the result is
     
  2. applying one of two functions to the result of step 1: taking e.g. , gives
     
  3. 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.