Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as \begin{equation} \int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx \end{equation} Here $a$ is positive real number and $f\left( {x|y} \right)$ is the Rice distribution $$ f\left( {x|y} \right) := \frac{x}{{{\sigma ^2}}}{e^{\left( { - \frac{{{x^2} + {y^2}}}{{{\sigma ^2}}}} \right)}}{I_0}\left( {\frac{{xy}}{{{\sigma ^2}}}} \right), $$ where ${I_0}$ is the modified Bessel function with order zero.
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$\begingroup$ I think you meant $a$ is a positive number. Are you looking for an asymptotic approximation or one for numerical work? $\endgroup$– A rural readerMar 15, 2021 at 3:35
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$\begingroup$ Also, might want to tack on a $dx$ to your integral. $\endgroup$– A rural readerMar 15, 2021 at 3:44
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