I need some help on a possible way to integrate this:

$$ \int_0^\infty{x^{m-1}\mathrm{e}^{-\lambda x}\left[\frac{\operatorname{K}_\nu\left(b\sqrt{\alpha+\beta x}\right)}{\left(b\sqrt{\alpha+\beta x}\right)^\nu}\right]}\mathrm{d}x $$ where $m$, $\lambda$, $b$, $\alpha$, $\beta$ are constants and $\operatorname{K}_\nu(\cdot)$ is the modified Bessel function of the second type.

  • 1
    $\begingroup$ What is the background of this question? What have you tried? I converted your picture into latex form; please check that I didn't make a mistake. $\endgroup$ – Eckhard Dec 25 '13 at 21:51
  • 2
    $\begingroup$ Did you mean $\alpha + \beta x$ instead of $\alpha+\beta \gamma$ in the denominator of the integrand? $\endgroup$ – Sasha Dec 25 '13 at 22:19
  • $\begingroup$ @Eckhard: this is related to comm. channels. I tried change of variable and assumed z=bsqrt(alpha+betax), but the integral got worse. $\endgroup$ – kazekage Dec 26 '13 at 5:28
  • $\begingroup$ @Sasha: Yes, thanks for the correction. $\endgroup$ – kazekage Dec 26 '13 at 5:29

Maple has this special case in terms of the Meijer G function: $$ \int _{0}^{1}\!{x}^{m-1}{{\rm K}_\nu\left(b\sqrt {\beta\,x}\right)} \left( b\sqrt {\beta\,x} \right) ^{-\nu}{dx}={2}^{-3+2\,m-\nu} \left( b\sqrt {\beta} \right) ^{-2\,m}{b}^{2}\beta\, G^{2, 1}_{1, 3}\left(\frac{\beta{b}^{2}}{4}\, \Big\vert\,^{0}_{m-1-\nu, m-1, -1}\right) $$ assuming $\nu$ is a positive integer.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.