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Consecutive nested integration (definite/indefinite)

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abouzar jafari
abouzar jafari on 28 Jan 2020
Answered: Alan Weiss on 29 Jan 2020
I am going to calculate a consecutive integration problem as I attached the picture here. I have tried to solve the integration using Mathematica and MATLAB even for the first two terms (F2, G2), but I did not find any meaningful results. Since the problem was solved for the first time in 1960 using a very simple computer, it can be solved using numerical integration. Actually, I have tried to do it using the code that I uploaded here, but the results obtained from the code does not match the solution. I hope someone can debug my code or give me some hints.
code:
clear all;
clc;
i = 1;
F{1} = @(x)(x./((1+x.^2).^2));
G{1} = @(x)((x.^3)./((1+x.^2).^2));
while i <= 20
i = i+1;
if mod(i, 2) == 0
% x is even [Sigma_y, x = 0]
FFny = @(x,y)(((y.*x.^2)./((y.^2+x.^2).^2)).*F{i-1}(x));
FGny = @(x,y)(((y.^3)./((y.^2+x.^2).^2)).*F{i-1}(x));
F{i} = @(y)integral(@(x)FFny(y,x),0,Inf);
G{i} = @(y)integral(@(x)FGny(y,x),0,Inf);
else
% x is odd [Sigma_x, y = 0]
FFnx = @(x,y)(((x.*y.^2)./((y.^2+x.^2).^2)).*F{i-1}(y));
FGnx = @(x,y)(((x.^3)./((y.^2+x.^2).^2)).*F{i-1}(y));
F{i} = @(x)integral(@(y)FFnx(y,x),0,Inf);
G{i} = @(x)integral(@(y)FGnx(y,x),0,Inf);
end
end

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Answers (1)

Alan Weiss
Alan Weiss on 29 Jan 2020
I would try to answer this question by not integrating from 0 to infinity, but instead from 0 to a large value, and then use an asymptotic expression for the remainder of the integration (from the large value to infinity). Some of these integrals converge very quickly, some not, so I suspect that a bit of pre-analysis before trying numerical integration would be very helpful.
Alan Weiss
MATLAB mathematical toolbox documentation

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