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2

Not sure what is going on with the different results of Integrate and NIntegrate. This does not mean that the analytic form of $C$ is erroneous. Note that plotting the likelihood function (using the expression of $C$ provided by Integrate and the parameter values you used) over a resticted range of $s$ (instead of $[0,1]$) clearly shows that the ...

3

The problem is numerically unstable for some parameter ranges. We shall show a simple example. Your normalized distribution is given by p[q_, n_, \[Mu]_, \[Nu]_, s_] := Exp[4 n s q] q^(4 n \[Nu] - 1) (1 - q)^(4 n \[Mu] - 1)/(Gamma[4 n \[Mu]] Gamma[ 4 n \[Nu]] Hypergeometric1F1Regularized[4 n \[Nu], 4 n (\[Mu] + \[Nu]), 4 n s]) Check ...

2

Well, I'll take a crack at it, although I think chris identified the nub of the problem in the first comment. We need to tighten up the language from the comments. First, write $$F(\theta)=F(\theta,k)=\sin \theta \int_{-L}^{+L}h(z)\,e^{-ikz\cos \theta} \,dz=\sin \theta \int_{-\infty}^{+\infty}\tilde{h}(z)\,e^{-ikz\cos \theta} \,dz$$ where ...

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