‘Pi over two’ conjecture

Statement

Suppose that x and y are positive reals, and ψ is an increasing function. Then, whenever the integral

makes sense, it is equal to π/2.

State of the art

• For ψ(t) = t, this is an easy excercise.
• For ψ(t) = √t, this is a much more difficult excercise.
• For ψ(t) = t^α (with 0 < α < 1) and many other complete Bernstein functions ψ, the result is proved in my recent preprint (arXiv:1006:0524).
• The conjecture can be verified numerically for various functions ψ with high precision. A sample Mathematica code for numerical experiments:

  psi[t_] = t^2 + Exp[-t] + Sin[Sqrt[t]];
  dpsi[t_] = D[psi[t], t];
  x = 3;
  y = 2;
  2/Pi NIntegrate[
    Hold[
     t^2 (x + y)/(x^2 + t^2)/(y^2 + t^2)
      Exp[1/Pi
         NIntegrate[
         (x/(x^2 + s^2) + y/(y^2 + s^2))
          Log[dpsi[t^2] (t^2 - s^2)/(psi[t^2] - psi[s^2])],
         {s, 0, Infinity}
         ]
       ]
     ],
    {t, 0, Infinity}, WorkingPrecision -> 50]
  

Origins

The conjecture was first stated in my preprint Spectral analysis of subordinate Brownian motions in half-line in 2010. It is closely related to the distribution of suprema of certain Lévy processes.

Contact

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