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For $\gamma>0,\delta>0$, How do I evaluate this integral?

$$ I=\int_0^H\frac{e^{i t x} \log\left(\frac{H}{H-x}\right) ^{\frac{1}{\gamma }-1} \left(\left(\frac{k}{H \log \left(\frac{H}{H-x}\right)}\right)^{-1/\gamma }+1\right)^{-\gamma -\delta -1}}{\gamma (H-x)}\,\mathrm{d}x,$$ with gratitude.

I tried the usual tricks, with no result so far.

BCLC
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Nero
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    Where did you catch this monster ? – Claude Leibovici Apr 05 '15 at 12:45
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    The theory of everything. – Relure Apr 05 '15 at 12:56
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    A probability distribution... long story. I transformed a Pareto IV distribution and trying to get the Characteristic Function. I can provide details if you like. – Nero Apr 05 '15 at 12:58
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    Wow... What is this? It only frigthens me... – Tolaso Apr 05 '15 at 13:02
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    A real problem monsters the masters. (to monster = to criticize someone severely or to treat someone very badly; http://dictionary.cambridge.org/dictionary/british/monster) – zoli Apr 05 '15 at 13:05
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    At least it looks a bit nicer if you substitute $x:=H(1-e^{-z})$ . :-) – user90369 Jun 14 '18 at 14:02
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    What is $t$?... – mathworker21 Jun 20 '18 at 05:40
  • Bounty: Intuition for Conditional Expectation (1. by linking bounty question here, this question gets attention because the bounty question is linked to this question. 2. by linking bounty question here, nero gets to see bounty question and so might answer bounty question) – BCLC Sep 02 '18 at 17:46
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    can you graph the function inside the integral for nominal values of $\gamma, \delta,$ etc ... perhaps the real part and imaginary parts separately. I'm a big believer in graphing functions as an aid in pondering their behavior – phdmba7of12 Jan 23 '19 at 15:12
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    By the u-sub u=(H-x)/H and then the realization that the non-phase portion is the derivative of (((k/H)/log(1/y))^(-1/gamma) +1)^(-gamma-delta) , you can them inductively reduce the problem to the Fourier transform of (((k/H)/log(1/y))^(-1/gamma) +1)^(-alpha) with respect to y where 0<alpha<1, but I don't see much more you can do with this. Is a power series solution acceptable? – Nicholas Parris Jul 19 '20 at 20:30
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    NNT/Nero, now that you haven't gotten responses or further details on maths se, why don't you go to maths overflow, stats se or operations research se? – BCLC Nov 19 '20 at 06:48
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    I can't even imagine how to solve it. Did you make any headway in these 6 years since you asked this here? – Loop Corrections Apr 30 '21 at 17:57
  • You can also ask on math tutors or physics stack exchange – Тyma Gaidash May 04 '21 at 19:13
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    @Nero You said you can add the details, if you do so this question will have improved right then. If you can do it now, we can reopen and see if someone's got anything. If you wish to weaken your integral and/or allow numerical approximations, you can do this as well, if you want a broader category of answers to be applicable. Thank you. – Sarvesh Ravichandran Iyer May 07 '21 at 18:16

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