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The marcum Q-function is defined by $$ Q_m(a,b) = \int^\infty_b x \left(\frac{x}{a}\right)^{m-1} \exp\left(-\frac{x^2+a^2}{2}\right) I_m\left(a x\right) \:\mathrm{d} x,$$ where $m\in\mathbb{N}$ , $b\in\mathbb{R}^+$ , $a\in\mathbb{R}^+$ , and $I_m(.)$ is the $m$-th order modified Bessel function of the first type. Is it possible to get the derivative of the Q-function with respect to $m$? |
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For large $x$, if $a>0$, $I_m$ behaves asymptotically like $I_m(ax)\approx e^{ax}/\sqrt{ax}$. Therefore for large $x$ the integrand will look like $x^{m-1/2}e^{-(x-a)^2/2}$. This dies off fast enough that the improper integral converges uniformly in $m$ and you can differentiate inside the integral. |
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