calculus and analysis - How to find a more precise value of integral?

Mathematica 11 produces

NIntegrate[ArcCot[x]*Sin[x]/(5/4 + Cos[x]), {x, 0, Infinity}, 
 Method -> "ExtrapolatingOscillatory", AccuracyGoal -> 10, 
 WorkingPrecision -> 20]
0.746733605129940959

This is 4.573.3 from Gradshteyn & Ryzhik. Up to G&R the result is

 N[Pi*Log[3/2/(1 + 1/2*Exp[-1])], 10]
   0.7433557514

The question arises: how to obtain it in Mathematica?

Answer

Here's an extrapolatory approach based on Richardson extrapolation of the integrals over each "period." (Richardson[] code from Anton Antonov's answer here):

Clear[Richardson]
Richardson[A_, n_, N_] := 
 Total@Table[(A[n + k]*(n + k)^N*If[OddQ[k + N], -1, 1])/(k! (N - k)!), {k, 0, N}]

(* the integral from 2 n Pi to 2 (n+1) Pi *)
Clear[aa];
aa[n_Integer /; n >= 0] := 
  aa[n] = NIntegrate[ArcCot[x]*Sin[x]/(5/4 + Cos[x]), {x, 2 n Pi, 2 (n + 1) Pi}, 

    WorkingPrecision -> 50];

We get around machine precision with 100 terms plus 6 extrapolation terms:

sf[n_] := Sum[aa[i], {i, 0, n}]; (* partial sum = integral over {0, 2 (n+1) Pi} *)
Richardson[sf, 100, 6]
% - Pi*Log[3/2/(1 + 1/2*Exp[-1])]
(*
  0.743355751361227474780881457479018423835    
  -2.97535713733265290609713*10^-16
*)

Without extrapolation:

sf[100]
% - Pi*Log[3/2/(1 + 1/2*Exp[-1])]
(*
  0.74207790286093203962531860751097243182349263171074
  -0.00127784850029573269127658323333660172377741318898
*)

Update

More extrapolatory terms yields an approximation accurate to machine precision with many fewer terms overall:

Richardson[sf, 1, 16]
% - Pi*Log[3/2/(1 + 1/2*Exp[-1])]
(*
  0.74335575136122767678505611095516483469029
  -9.553153907978914419885698*10^-17
*)

Comments

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Adding a thick curve to a regionplot

Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],

Blog

Search This Blog