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Solving a certain inverse problem reduces to solving the following elementary equation: $$ (A^2+x^2)^sx = C $$ Here

  • $x$ is the unknown. We are interested in $x > 0$.
  • $s \in \mathbb{R}$ is a real number.
  • $A > 0$ is a fixed real number.
  • $C > 0$ is a fixed number, small enough that the equation has at least one solution.

It turns out that depending on the value of $s$ the equation has one or at most two solutions. The left side is strictly increasing when $s \ge -1/2$. It is strictly increasing up to a given point, after which it is strictly decreasing, when $s < -1/2$.

For what values of $s$ does the equation have a closed form solution?

I know of the following cases:

  • $s = -1$ second order polynomial
  • $s = -1/2$ second order polynomial
  • $s = 0$ trivial
  • $s = 1/2$ fourth order polynomial with special form
  • $s = 1$ third order polynomial, which I did not bother to solve

Are there are more, and is it possible to verify that all have been found?

Note that the equation is polynomial if $s$ is rational, but not all polynomial equations have explicit solutions.

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