Consecutive nested integration (definite/indefinite)

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

Answer 1

[Alan Weiss]

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

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