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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:



Mathematica graphics




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 *)

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