2
$\begingroup$

I'm trying to model an experiment for a paper which includes an integral that should be no problem for mathematica but it refuses to integrate it. The integral I need to evaluate is as follows

G[q_, t_] := (E^(2 I Sqrt[676000000 \[Pi]^2 -q^2] (23/1000000 + (1 - Cos[75000 t])/20000))q^3 ((8.46*^11 + 2.63*^11 I) Sqrt[1.88*^-13 - q^2] - Sqrt[(0.16 + 0.049 I) - q^2]))/(Sqrt[ 676000000 \[Pi]^2 - q^2] ((8.46*^11 + 2.63*^11 I) Sqrt[1.88*^-13 - q^2] + Sqrt[(0.15 + 0.049 I) - q^2]));

enter image description here

As you can see, it's a function of both q and t and I need to integrate it over q while keeping t constant as I have to integrate over t later on in the simulation. I have no idea why it won't integrate, please help!

Supplementary Information - this is me trying to solve the problem

The function G here is made up of two functions, rp, and another called Phi_{PDM}.

Phi_{PDM} integrated over q is no problem, it's also where the t dependence comes from, I've included here for completeness.

enter image description here

The problem lies with the function rp as I cannot get mathematica to integrate this function over q

enter image description here

enter image description here

As you can see, there is a singularity at small values of q which is probably whats causing the problem and I tried to mitigate this with assuming q is real and/or is above 10 but this still does not help. Here is the code for the equations:

Phi[q_, t_] := (q^3/(Sqrt[(2*\[Pi]*13000)^2 - q^2]))*Exp[I*2* (Sqrt[(2*\[Pi]*13000)^2 - q^2])*(23*^-6 + 50*^-6*(1 - Cos[75000*t]))]; rp[q_] := ((8.46*^11 + 2.63*^11 I) Sqrt[1.88*^-13 - q^2] - Sqrt[(0.16 + 0.049 I) - q^2])/((8.47*^11 + 2.63`*^11 I) Sqrt[ 1.88*^-13 - q^2] + Sqrt[(0.16 + 0.049 I) - q^2])

I'm not very good at mathematica and I'm certainly not the savviest when it comes to mathematics in general so any ideas regarding this, or indeed a method of solving G that I'm just not seeing due to me being a novice, would massively appreciate!

Improve this question
$\endgroup$
14
  • $\begingroup$ Welcome to the Mathematica Stack Exchange. Have you tried NIntegrate? $\endgroup$ Commented Aug 26, 2022 at 17:50
  • $\begingroup$ Hey there, thanks for the welcome. As I said in the post, NIntegrate is useless for my purposes, I must solve it with the Integrate function $\endgroup$ Commented Aug 26, 2022 at 17:52
  • 1
    $\begingroup$ It is not quite understandable: You have to integrate a numerical function und must avoid NIntegrate ? $\endgroup$ Commented Aug 26, 2022 at 19:07
  • $\begingroup$ Well I have isolated the problem I'm having later in the code with this expression. For my simulation there is also a time dependent variable "t" included in the integral leaving me with an expression to integrate over "q" while leaving "t" constant. Since I know it's this particular part of of the expression that's giving me trouble, I thought I would simplify but if you think it makes more sense to include the full expression then I shall do so! $\endgroup$ Commented Aug 26, 2022 at 19:21
  • 1
    $\begingroup$ @Koji Please show the full expression. Knowing rp[q,t] you can calculate int[t_ NumericQ]:=NIntegrate[rp[q,t],{q,0,20}] depending on t. $\endgroup$ Commented Aug 26, 2022 at 19:32

1 Answer 1

Reset to default
3
$\begingroup$

I solved the issue and thought it instructive to post the solution:

A contributor in the comments mentioned using NumericQ with NIntegrate, a function I had never previously come across. Turn out you can attach this to variable in an equation and it acts as a placeholder until such time that there is a number for that variable that then allows you to solve the problem numerically

rp[q_] := ((8.46*^11 + 2.63*^11 I) Sqrt[1.88*^-13 - q^2] - Sqrt[(0.16 + 0.049 I) - q^2])/((8.47*^11 + 2.63`*^11 I) Sqrt[ 1.88*^-13 - q^2] + Sqrt[(0.16 + 0.049 I) - q^2]) \[Phi][\[Omega]_, q_, t_] := (q^3/(Sqrt[(2*\[Pi]*\[Omega])^2 - q^2]))* Exp[I*2*(Sqrt[(2*\[Pi]*\[Omega])^2 - q^2])*z[t]]; G[\[Omega]_, t_?NumericQ] := NIntegrate[rp[\[Omega], q]*\[Phi][\[Omega], q, t], {q, 0, 2000000}] array = ConstantArray[0, 50]; For[i = 1, i < 51, i++, array[[i]] = NIntegrate[Rationalize[FullSimplify[\[Alpha]tip[[i]]/( 1 - \[Alpha]tip[[i]]*G[\[Nu][[i]], t])]]*Exp[I*5*75000*t], {q,0, 2000000}, {t, 0, 2 \[Pi]/75000}, AccuracyGoal -> 5]; ]

enter image description here

Improve this answer
$\endgroup$

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.