Is there any method to calculate above integral? I need to know answer for different $q$'s. $J_1$ is Bessel function.
$\partial_\rho\bigg (\int_0^{\infty} \frac{\rho_0^2 J_1(q \rho_0)}{(a^2+(\rho_0-\rho)^2)^{\frac{5}{2}}} d\rho_0 \bigg)$
2026-03-26 07:35:29.1774510529
$\partial_\rho\bigg (\int_0^{\infty} \frac{\rho_0^2 J_1(q \rho_0)}{(a^2+(\rho_0-\rho)^2)^{\frac{5}{2}}} d\rho_0 \bigg)$
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I don't think there is a closed form for the indefinite integral, but the definite integral from $\rho=0$ to $\infty$ (for $a > 0$ and $q > 0$) is $$\frac{q}{6 a^2} \left( aq{{I}_{0}\left(\frac{aq}{2}\right)}{{ K}_{1} \left(\frac{aq}{2}\right)}-aq{{ I}_{1}\left(\frac{aq}2\right)}{{ K}_{0} \left(\frac{aq}2\right)}-2\,{{ I}_{1}\left(\frac{aq}2\right)}{{ K}_{1} \left(\frac{aq}2\right)} \right) $$ where $I_j$ and $K_j$ are modified Bessel functions of the first and second kinds.