Principal Value in NIntegrate. Integrand with multiple poles

I want to evaluate the numerical integral:

I1[y_] := NIntegrate[PolyGamma[(3*((0.61^2) - x (1 - x)*(y)^2))/(4*0.001) + 1], {x, 0, 1}]

The plot of the function defined by the integral above is: The result is highly oscillatory for y>2*0.61. This happens because the PolyGamma function has poles if the argument is a negative integer or zero. My idea is to use Principal Value and exclude the points which the argument is negative and integer.

I reckon that the best approach is:

I2[y_] := NIntegrate[PolyGamma[(3*((0.61^2) - x (1 - x)*(y)^2))/(4*0.001) + 1], {x, 0, 1}, Method ->PrincipalValue, Exclusions -> {(3*((0.61^2) - x (1 - x)*(y)^2))/(4*0.001) + 1 == n}]

But I don’t know how to define "n" as a negative integer. I have two questions:

1. How can I define n as a negative integer and use it to exclude the problematic points?

2. There is another way to compute the principal value of this integral?

Mathematica Asked by Victor Alencar on December 28, 2020

Another way. Using primary definition of PolyGamma function as a series:

Integrate[(z/(k (k + z)) - 1/z - EulerGamma) /.
z -> Rationalize[(3*((0.61^2) - x (1 - x)*(y)^2))/(4*0.001) + 1,
0], {x, 0, 1}, Assumptions -> y >= 0]

(*-EulerGamma + 1/k - (
4 ArcTan[(50 Sqrt y)/Sqrt[11203 + 40 k - 7500 y^2]])/(
5 Sqrt y Sqrt[11203 + 40 k - 7500 y^2]) - (
4 ArcTan[(50 y)/Sqrt[11203/3 - 2500 y^2]])/(
5 y Sqrt[33609 - 22500 y^2])*)

and we have:

F[y_, M_] := -EulerGamma +
Sum[1/k - (4 ArcTan[(50 Sqrt y)/Sqrt[11203 + 40 k - 7500 y^2]])/(5 Sqrt y
Sqrt[11203 + 40 k - 7500 y^2]), {k, 1, M}]
- (4 ArcTan[(50 y)/Sqrt[11203/3 - 2500 y^2]])/(5 y Sqrt[33609 - 22500 y^2])

Plot[F[y, 2000] // Re, {y, 0, 2}, PlotRange -> {Automatic, {2, 6}}] Plot[F[y, 5000] // Re, {y, 0, 2},
PlotRange -> {{1.225, 1.5}, {2, 6}}](*In range 1.225 < x < 1.5*) Answered by Mariusz Iwaniuk on December 28, 2020

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