list manipulation - Failure in integrating from an interpolating function!

Consider a list like below:

ttable={{0, 2.6596 - 66.137 I}, {1/9, 2.45339 - 65.3148 I}, {2/9, 
  1.82053 - 62.8922 I}, {1/3, 0.720006 - 58.9982 I}, {4/
  9, -0.911205 - 53.8382 I}, {5/9, -3.15056 - 47.6797 I}, {2/
  3, -6.08057 - 40.8346 I}, {7/9, -9.77826 - 33.6382 I}, {8/
  9, -14.3047 - 26.4282 I}, {1, -19.6947 - 19.5216 I}}`

by defining "f" as interpolating function we'll have:

f = Interpolation[ttable]

now we define ff as the following integration:

ff[\[Xi]_]:=NIntegrate[(f[rr]) Cos[rr \[Xi]], {rr, 0, 1}]

the first problem is that this function can not be calculated and I have no idea why the followoing error happens:

In[87]:= ff[.1]
During evaluation of In[87]:= NIntegrate::inumr: The integrand Cos[rr \[Xi]] InterpolatingFunction[{{0.,1.}},{4,15,0,{10},{4},0,0,0,0,Automatic,{},{},False},<<1>>,{
Developer`PackedArrayForm,{0,<<10>>},{2.6596 -66.137 I,<<8>>,-19.6947-19.5216 I}},{Automatic}][rr] has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,0.111111}}. >>
Out[87]= NIntegrate[f[rr] Cos[rr \[Xi]],{rr,0,1}]

The other thing should be mentioned is that I have to use ff in integrand of another integration like below:

NIntegrate[ff[\[Xi]] * \[Xi],{\[Xi],0,3}]

I've manipulated the former equations in many ways but none of them made an accurate output for the last integration, so I would be greatly thankful if somebody out there could help me!

Answer

The most important information is that you used your function ff inside another NIntegrate, because this is the source of confusion. What you have to know is that NIntegrate doesn't start right away with the numerical calculation when you call

NIntegrate[ff[ξ]*ξ, {ξ, 0, 3}]

It will try to do some analysis of your integrand and most likely, it will try to evaluate ff[ξ] without putting in numbers. And what happens then? Right, you call the NIntegrate of ff without proper numerical value of ξ:

The solution is pretty simple: Change your definition of ff so that it only calls its NIntegrate body when the argument is indeed numeric:

ClearAll[ff];
ff[ξ_?NumericQ] := NIntegrate[(f[rr]) Cos[rr ξ], {rr, 0, 1}]
NIntegrate[ff[ξ]*ξ, {ξ, 0, 3}]

(* 4.28747 - 124.522 I *)

Comments

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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]],

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