# Solve a Cauchy integral if the singular node is an endpoint?

GROUPS:
 I have one definite integral that needs to be evaluated from -1 to 1 in the Cauchy principal value sense. Math programs only let me put the singular points between the integration limits , but my singular point is -1.The integral is complex,it has a strong singularity on -1, but how can I integrate from -1 to 1?Here is the function for simple copy and paste: -(((1 - xi)*xi*Sqrt[(0.056249999999999994*(1 - xi) + 0.10625000000000001*xi - 0.049999999999999996*(1 + xi))^2]* (0. + ((0.056249999999999994*(1 - xi) + 0.10625000000000001*xi - 0.049999999999999996*(1 + xi))*(-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))* (1.3 + 4*(BesselK[0, 628.318530717959*Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2]] + (0.003183098861837905*(-(0.0015915494309189525/Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2]) + BesselK[1, 628.318530717959*Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2]]))/ Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2])))/(Sqrt[(0.056249999999999994*(1 - xi) + 0.10625000000000001*xi - 0.049999999999999996*(1 + xi))^2]* Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2]) + (0.*(-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))* (0.7 + 1256.637061435918*Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2]* BesselK[1, 628.318530717959*Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2]] + 8*(BesselK[0, 628.318530717959*Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2]] + (0.003183098861837905*(-(0.0015915494309189525/Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2]) + BesselK[1, 628.318530717959*Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2]]))/ Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2])))/(Sqrt[(0.056249999999999994*(1 - xi) + 0.10625000000000001*xi - 0.049999999999999996*(1 + xi))^2]* Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2])))/(8*Pi*Sqrt[0. + (-0.1 - 0.05*(1 - xi)*xi + 0.10625*(1 - xi)*(1 + xi) + 0.05625*xi*(1 + xi))^2])) Could mathematica solve this integral since the singular point is on the endpoint of the integration?
1 year ago
5 Replies
 Does anyone have a hint on how to solve the above integral ?
1 year ago
 How do you define this integral? That is to say, what normalization do you have in mind to remove the singularity.