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I have an integral which doesn't give a closed definite expression. The command

Integrate[x DiracDelta[r x - y] Exp[1/g^2 {Cos[x2 - x] + Cos[x2] + Cos[y+x2]}], {x, 0, 2 Pi}, {y, 0, 2 Pi}, {x2, -Pi, Pi}]

returns the expression itself. Therefore I evaluate this numerically using NIntegrate in order to understand the behaviour of the integral with $r$ and $g^2$.

In this situation my preferred way is to to obtain plots of the integral with respect to $g^2$ for a fixed $r$, and also with $r$ for a fixed $g^2$. How do I combine Plot and NIntegrate together and get a plot for the same over some values for $r$ and $g^2$? In other words, I would need to iterate NIntegrate, and construct a list of points so as to plot on the graph. How do I achieve that?

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  • $\begingroup$ You may define a function f[r_?NumericQ, g_?NumericQ]:=NIntegrate[...]. $\endgroup$
    – Alx
    Feb 29, 2020 at 0:48
  • $\begingroup$ @Alx Can you make it an answer? How would I generate the plot afterwards? $\endgroup$ Feb 29, 2020 at 1:22
  • $\begingroup$ Plot3D[f[r,g],{r,...},{g,...}], you need to put in ranges for r and g. $\endgroup$
    – Alx
    Feb 29, 2020 at 1:36

2 Answers 2

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Progress can be made by integrating over y only (with the correction that {…} be replaced by (…)).

Integrate[x DiracDelta[r x - y] Exp[1/g^2 (Cos[x2 - x] + Cos[x2] + Cos[y + x2])], 
    {y, 0, 2 Pi}, Assumptions -> r > 0 && 0 < x < 2 Pi]
(* E^((Cos[x - x2] + Cos[x2] + Cos[r x + x2])/g^2) x HeavisideTheta[2 Pi - r x] *)

In contrast, if r < 0, the result is 0, as one would expect.

The remaining integrals take the form,

Integrate[E^((Cos[x - x2] + Cos[x2] + Cos[r x + x2])/g^2)
    x HeavisideTheta[2 Pi - r x], {x, 0, 2 Pi}, {x2, -Pi, Pi}]

Edit: which can be simplified further to

Integrate[E^((Cos[x - x2] + Cos[x2] + Cos[r x + x2])/g^2) x, 
    {x, 0, 2 Pi Min[1, 1/r]}, {x2, -Pi, Pi}

Now, if g^2 is much greater than 1, then the remaining integrations can be performed with the exponential ignored.

Integrate[x, {x, 0, 2 Pi Min[1, 1/r]}, {x2, -Pi, Pi}, Assumptions -> r > 0]
(* (4 Pi^3 Min[1, 1/r]^2 *)

For g^2 not large, the remaining integrals must be performed numerically along the lines suggested by Alx in a comment above, but a 2D plot is slow.

f[r_?NumericQ, g_?NumericQ] := NIntegrate[E^((Cos[x - x2] + Cos[x2] + Cos[r x + x2])/g^2) x, 
    {x, 0, 2 Pi Min[1, 1/r]}, {x2, -Pi, Pi}, 
    Method -> {Automatic, "SymbolicProcessing" -> False}]

A 1D plot for g == 1 is

Plot[f[r, 1], {r, 0, 4}, MaxRecursion -> 2, ImageSize -> Large, 
    AxesLabel -> {r, None}, LabelStyle -> {15, Bold, Black}]

enter image description here

For a small value of g, the curve is somewhat different.

LogPlot[f[r, 1/10], {r, 0, 4}, MaxRecursion -> 3, ImageSize -> Large, 
    AxesLabel -> {r, None}, LabelStyle -> {15, Bold, Black}, PlotRange -> All]

enter image description here

Peaks lie at r = 1 and very close to r = 0.

Addendum

The further simplification identified by Andreas (+1), equivalent to

f[r_?NumericQ, g_?NumericQ] := 2 Pi NIntegrate[BesselI[0, g^-2 
    Sqrt[3 + 2 Cos[x] + 2 Cos[r x] + 2 Cos[(1 + r) x]]] x, {x, 0, 2 Pi Min[1, 1/r]}, 
    Method -> {Automatic, "SymbolicProcessing" -> False}]

reproduces the plots above about an order of magnitude faster. For completeness, we provide a plot of the values at the two peak values as functions of g (rescaled by Exp[-2.94 g^-2] to keep the curves readable.

LogPlot[{f[.02, g] Exp[-2.94 g^-2], f[1, g] Exp[-2.94 g^-2]}, {g, 1/10, 3},
    ImageSize -> Large, AxesLabel -> {g, None}, LabelStyle -> {15, Bold, Black}, 
    PlotRange -> {{0, 3}, Automatic}]

The blue and orange curves correspond to r = 0.02 and r = 1, respectively.

enter image description here

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  • $\begingroup$ For my case g^2 is not large, so I won't be able to use this. Thanks though. $\endgroup$ Feb 29, 2020 at 5:59
  • $\begingroup$ Even so, doing the y integration symbolically speeds the subsequent numerical computations and may provide insight. I just added a plot for g == 1 to illustrate the plotting process. $\endgroup$
    – bbgodfrey
    Feb 29, 2020 at 6:09
  • $\begingroup$ I see your point. Let me see if I can use this. $\endgroup$ Feb 29, 2020 at 7:49
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if the range of the x2 integration may be shifted to {0,2π} then it can be done and plotted like:

g = 2.; Plot[{NIntegrate[
   E^((Cos[x - x2] + Cos[x2] + Cos[r x + x2])/g^2) x HeavisideTheta[2 Pi - r x], {x, 0, 2 Pi}, {x2, 0, 2 Pi}], 
  2 Pi  NIntegrate [x BesselI[0, 1/g^2 \[Sqrt](3 + 2 Cos[x] + 2 Cos[r x] + 
           2 Cos[(1 + r) x])] HeavisideTheta[2 Pi - r x], {x, 0, 
      2 Pi}] + 10}, {r, 0, 4}]

because

Integrate[Exp[a Cos[x]] Exp[c Sin[x]], {x, 0, 2 Pi}]
is (* 2 Pi BesselI[0,Sqrt[a^2+c^2]]*)

Above is valid if you shift the integration range by any s (for instance s=-Pi):

Integrate[Exp[a Cos[x]] Exp[c Sin[x]], {x, s, 2 Pi+s}]
is (* 2 Pi BesselI[0,Sqrt[a^2+c^2]]*)
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  • $\begingroup$ Thanks, let me see if this works out. $\endgroup$ Mar 1, 2020 at 10:04

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