Approximation for the Marcum Q-function

Question

I want to approximate the Marcum Q-function $Q_{m}(a,b)$ when $b$ goes to zero, where the Marcum Q-function $Q_{m}(a,b)$ is defined in an integral form, shown in https://en.wikipedia.org/wiki/Marcum_Q-function

Answer 1

Assume that all three parameters are positive. We know that the integral over $[0, \infty)$ is $1$. The integral over $[0, b]$ can be evaluated termwise: $$e^{-x^2/2} I_{m - 1}(a x) = \frac {\left( \frac {a x} 2 \right)^{m - 1}} {(m - 1)!} \sum_{n \geq 0} \frac {(-1)^n} {n!} \hspace {1px} {_1 \hspace {-1.5px} F_1} {\left( -n; m; \frac {a^2} 2 \right)} \left( \frac {x^2} {2} \right)^{\! n}, \\ Q_m(a, b) = 1 - \frac {e^{-a^2/2}} {(m - 1)!} \sum_{n \geq 0} \frac {(-1)^n} {(n + m) n!} \hspace {1px} {_1 \hspace {-1.5px} F_1} {\left( -n; m; \frac {a^2} 2 \right)} \left( \frac {b^2} 2 \right)^{\! n + m}.$$