I am working on a university project at the moment and at some point I needed to calculate the intergral of the following function (Please refer to "Bakthiari et al. - Analysis of radiation from an open-ended coaxial line into stratified dielectrics", if you are interested in the nature of the problem)
$y_s=\int_0^\infty \frac{[J_0(k_0\zeta b)-J_0(k_0\zeta a)]^2}{\zeta}F(\zeta)d\zeta$
where $J_0(x)$ is the zeroth order Bessel function of first kind and $F(\zeta)$ is a complex non-oscillatory function with singularities. $k_0, \zeta, b$ and $a$ are all real constants.
$F(\zeta)=\frac{1}{\sqrt{\epsilon_{r1}-\zeta^2}} [\frac{\kappa_1+j \tan{(k_0d_1\sqrt{\epsilon_{r1}-\zeta^2})}}{1+j\kappa_1tan(k_0d_1\sqrt{\epsilon_{r1}-\zeta^2})}]$ and $\kappa_1=\frac{\epsilon_{r_2}}{\epsilon_{r_1}} \frac{\sqrt{\epsilon_{r1}-\zeta^2}}{\sqrt{\epsilon_{r2}-\zeta^2}}$, where $\epsilon_{r1}, \epsilon_{r2}$ and $d_1$ are real constants as well. $j$ is the imaginary unit.
As you can see this integral is rather complicated and when I used a computer algebra system to calculate this integral the calculation took some time(~60s). Unfortunately, I need to calculate the inverse of this function with respect to one of the constants and, therefore, I need to be a able to quickly compute this intergral.
I have tried to perform this integration using MATLAB, unfortunately the paper that I have referred to above also states that due to poles of the real part of the integrand the intergration needs to be carried out using contour integral techniques. Does anyone have an idea how to convert this integral into a numerically computable integral?
Thanks, Max