I need to compute the Hankel transform of an algebraic function that is not found in standard Hankel transform integral tables. I tried to compute it exactly with Maple or Mathematica and also rubi, but all fail to deliver an exact solution. The numerical solution however is of course possible but not the anticipated solution.
The integral is $$I(r;\tau,\epsilon):=\int_0^{\infty} dk k J_0(k r) f(k;\tau,\epsilon) $$ which is the Hankel transform of the function $$ f(k;\tau,\epsilon):=\left(\frac{k^2 \tau \epsilon +\epsilon }{k^4 \tau \epsilon +k^2 (\tau +\epsilon +1)+1}\right)^p$$ where
- $p=1$ or $p=1/2$ (the square root of the given algebraic function), and
- the parameters $\epsilon$ and $\tau$ are reals with $\tau\geq 0$ and $\epsilon\geq 0$.
The function $f(k;\tau,\epsilon)$ is an approximation to the exact function $$ f_{ex}(k;\tau,\epsilon):=\left(\frac{\epsilon \tau}{1+\tau + k^2 \epsilon \tau - e^{-k^2 \tau} I_0(k^2 \tau)}\right)^p $$ from which we obtain $f(k;\tau,\epsilon)$ by using the Pade approximation for $e^{-k^2 \tau} I_0(k^2 \tau)\approx 1/(1+k^2 \tau)$. Of course the Hankel transform of the exact function $f_{ex}(k;\tau,\epsilon)$ is of similar interest.
Update: Utilizing partial fraction decomposition on the Hankel transform function yields $$ f(k;\tau,\epsilon)=\left(\frac{\alpha_+^2}{1+\beta_-^2 k^2}+\frac{\alpha_-^2}{1+\beta_+^2 k^2} \right)^p $$ with with four coefficients $\alpha_\pm(\epsilon,\tau)$ and $\beta_\pm(\epsilon,\tau)$.
With this we can calculate the $p=1$ case to $$I(r;\tau,\epsilon)=\left(\frac{\alpha_+}{\beta_-}\right)^2K_0(r/\beta_-)+\left(\frac{\alpha_-}{\beta_+}\right)^2K_0(r/\beta_+)$$
The exact integral of the $p=1/2$ case is still unknown. Any tips and help would be of interest.