I've got the following problem I'm working on which is related to some of my research.

I am trying to solve the following equation for the function $f$.

$$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \alpha} \right)} = \int_0^t \frac{f\left(x, s\right)}{t - s}ds$$ where $0 \le \alpha \le 1$ and $\beta=\alpha \Gamma \left(\alpha+\frac{1}{2}\right)^2$ are some non negative constants.

I have tried to take Laplace and Fourier transforms of the left hand side, but to no avail. Is there any other strategy to obtain $f$ ? I am stuck here, any help would be greatly appreciated. Thanks.

  • $\begingroup$ Isn't there a problem with convergence of the integral at s=t? $\endgroup$ Apr 10, 2016 at 17:22
  • $\begingroup$ @MichaelRenardy ... That right-hand side is known as a "singular integral". There is a vast literature on that. $\endgroup$ Apr 10, 2016 at 17:27
  • 2
    $\begingroup$ It is a one-sided integral, so it cannot be interpreted as a Cauchy principal value. What am I missing? $\endgroup$ Apr 10, 2016 at 17:54


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