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I have a pretty ferocious integral to solve, and since it doesn't seem I'll be able to do much analytically, I've taken to Mathematica to get some information. Mainly, I want to see if there are any constant terms in the following integral:

$$ I = \int_{r}^{\infty} r_0^{-5/2} W_{-i\alpha'/2, \nu}\left(\frac{-ir_0^2\omega'}{L}\right) W_{-i\alpha/2, \nu}\left(\frac{-ir_0^2\omega}{L}\right) J_{\nu}\left(-i\hat{k}r_0\right) dr_0$$

Where $W$ is WhittakerW, $J$ is BesselJ, and everything besides $r/r_0$ is a constant.

So far, I've simply been doing NIntegrate and breaking the integral up over several pieces. I plotted the function on a log plot, and it seems after about $r \approx 10$ it drops off significantly; as such, the integral from 10 to infinity seems to be zero. After this, I've just plotted the integral and tried to look at dependence using LogPlots. Does anyone have a better method to deal with Numerical Integrals like this to get a bit more information?

Here is my mathematica code: 

g[r_, α_, v_, w_] := WhittakerW[-I*α/2, v, -I*r^2*w]

Plot[Log[(Re/Im)[
 r^(-5/2)*g[r, 10, 2.1, .01 ]*g[r, 15, 2.1, .01]*
BesselJ[3.2, 5*r]]], {r, 1, 100}, PlotLabel -> "Log(I2) vs r0"]

 % see that the function being integrated goes to zero for both Real and Imaginary parts


 I2bh[r_? NumericQ, v_] := 
 Re[NIntegrate[
 r0^(-5/2)*g[r0, 10, 2.1, .01 ]*g[r0, 15, 2.1, .01]*
  BesselJ[3.2, 5*r0], {r0, r, 15}, WorkingPrecision -> 15]]

I've done some more plotting and probing of the features of the integral, but for the purposes of this question I suppose what I have here is all I'm asking about

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    $\begingroup$ Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Read the faq! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! $\endgroup$
    – Michael E2
    Commented Aug 17, 2015 at 14:06
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    $\begingroup$ People here generally like users to post code as Mathematica code instead of images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find this this meta Q&A helpful $\endgroup$
    – Michael E2
    Commented Aug 17, 2015 at 14:07
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    $\begingroup$ Maybe my brain is blanking out, but could you explain what you mean be a constant term? You want to express $I$ as $I(r) = I_0 + f(r)$, where $f(r)$ meets some condition? $\endgroup$
    – Michael E2
    Commented Aug 17, 2015 at 14:21
  • $\begingroup$ Sorry about that - see my revised post. And indeed I want to express $I(r) = I_0 + f(r)$ where $I_0$ is a constant. Analytically doing this integral around $r \approx 0$ I don't see such a term, but this simplification probably isn't precise. $\endgroup$
    – Schwinger
    Commented Aug 17, 2015 at 18:39
  • $\begingroup$ You can try asking on math.stackexchange. You might get some pointers on how to proceed. Some of the guys on there are real integral wizards. See e.g. math.stackexchange.com/questions/562694/… $\endgroup$
    – QuantumDot
    Commented Aug 17, 2015 at 18:52

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