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Question: Please help me calculate the integral with singular points

Question:Please help me calculate the integral with singular points

DimaBrt 100 Maple 18
integration cauchyprincipalvalue numeric
November 09 2021
1

Hello. Please help me. I need to calculate the integral (3). This integral has many singular points at which there is convergence in the sense of the principal Cauchy value. The Maple integral itself does not count. I don't understand how to find automatically all the singular points on the integration area. Then, perhaps, it would be possible to split the integral into the sum of integrals by regions, as I roughly wrote in the picture. I want to automate this process, because in fact it is necessary to calculate many integrals of the form (4), where f(x,y) are arbitrary functions that can oscillate strongly, so I don't want to write banal quadrature formulas. I would like to use the means of Maple, since the accuracy will be greater and faster, but we need to somehow bypass the special points. I will be glad of any help. Thank you very much


> 

restart
> 

r1 := 1:

1/1000000

 (1)
> 

F1 := 1/(Zp*sqrt(k^2-x^2)*sin(y)+omega*rho1);

1/(46715093.93*(-x^2+1)^(1/2)*sin(y)+1485000)

 (2)
> 

Int(F1, x = -k+epsilon .. k-epsilon, y = 0 .. 2*Pi);

Int(1/(46715093.93*(-x^2+1)^(1/2)*sin(y)+1485000), x = -999999/1000000 .. 999999/1000000, y = 0 .. 2*Pi)

 (3)
> 

F2 := F1*f(x, y);

f(x, y)/(46715093.93*(-x^2+1)^(1/2)*sin(y)+1485000)

 (4)
> 

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