0
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I am looking for an analytic inverse function for

u0[alpha_] := (p0 (1/2 (((m1 - m2) (m1 + m2))/p0 + p0) + ((1 - 
        2 alpha)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/(
     2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
        1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + 
           p0)^2)) - \[Sqrt](((1 - 
           2 alpha)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/(
        4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
           1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + 
              p0)^2)^2) + ((1 - 2 alpha)^2 m1^4 + 
         m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/(
        p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
         1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + 
            p0)^2))))/(p0 + ((1 - 
      2 alpha)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/(
   2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
      1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + 
         p0)^2)) - \[Sqrt](((1 - 
         2 alpha)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/(
      4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
         1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + 
            p0)^2)^2) + ((1 - 2 alpha)^2 m1^4 + 
       m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/(
      p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
       1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + 
          p0)^2)) + (1 - 
      2 alpha) \[Sqrt](-m1^2 + (-(((1 - 
             2 alpha)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/(
          2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
             1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + 
                p0)^2))) + \[Sqrt](((1 - 
               2 alpha)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/(
            4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
               1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + 
                  p0)^2)^2) + ((1 - 2 alpha)^2 m1^4 + 
             m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/(
            p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
             1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + 
                p0)^2)))^2));

With p0 > 0, 0 < m1 < p0/2 , 0 < m2 < p0/2. I have already tried to invert this function in full generality with the help of Gröbner basis, with InverseFunction ad with Solve, but nothing succeeded so far. For InverseFunction, i used

Assuming[p0 > 0 && 0 < m1 < 1/2 && 0 < m2 < 1/2 && 0 <= xi <= 1 && 
  0 <= # <= 1, 
 Simplify[InverseFunction[(p0 (1/
           2 (((m1 - m2) (m1 + m2))/p0 + p0) + ((1 - 
             2 #)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/(
          2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
             1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + 
                p0)^2)) - \[Sqrt](((1 - 
                2 #)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/(
             4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
                1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + 
                   p0)^2)^2) + ((1 - 2 #)^2 m1^4 + 
              m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/(
             p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
              1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + 
                 p0)^2))))/(p0 + ((1 - 
           2 #)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/(
        2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
           1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + 
              p0)^2)) - \[Sqrt](((1 - 
              2 #)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/(
           4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 

              1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + 
                 p0)^2)^2) + ((1 - 2 #)^2 m1^4 + 
            m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/(
           p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
            1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + 
               p0)^2)) + (1 - 
           2 #)*\[Sqrt](-m1^2 + (-(((1 - 
                  2 #)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/(
               2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
                  1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + 
                    p0)^2))) + \[Sqrt](((1 - 
                    2 #)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/(
                 4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
                    1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + 
                    p0)^2)^2) + ((1 - 2 #)^2 m1^4 + 
                  m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + 
                    p0)^2)/(
                 p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 
                  1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + 
                    p0)^2)))^2)) &][u0]]]

But this does not seem to produce any results at least on my machine. Do you have any suggestions how to tackle this issue? Please keep in mind that i am explicitly looking for an analytic function alpha[u0] in the end.

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  • $\begingroup$ Would a series approximation suffice? $\endgroup$ – rhermans Jul 18 at 11:18
  • $\begingroup$ I am pretty sure that an approximation is not even needed. You can calculate the inverse function for a given set of constants like p0 = 1 , m1 = 1/10, m2 = 1/10 and you will see that this expression is very simple, just a square root appears. You can generalize this even further by assuming p0 =1 and m1 = m2 and the result will still be very simple. $\endgroup$ – RealDestructor Jul 18 at 11:21
  • $\begingroup$ Not invertible: m1 = 1; m2 = 2; p0 = 3; u0[0] == u0[1] returns True. $\endgroup$ – Henrik Schumacher Jul 18 at 12:09
  • $\begingroup$ This case is forbidden by physics, check out the assumptions. In my restricted area, this function should be bijective. $\endgroup$ – RealDestructor Jul 18 at 12:53

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