**Example 1.**

$$\mathcal{H}(f,t) = \oint\_{-1}^{1} \frac{\sin x}{x-t} dx, \quad \mathcal{H}^1(f,t) = \oint\_{-1}^{1} \frac{\sin x}{(x-t)^2} dx, \quad t = 0.1.$$

*Here f* ∈ *C*∞ *and as we can see the performance of the quadrature rules improves keeping m fixed and increasing the values of s. An empty cell means that there is no improvement in the computation. In particular as we can see in Tables 1–2, the machine precision is attained for m* = 128 *and s* = 16 *as well as for m* = 64 *and s* = 32*.*

**Table 1.** Example 1a: .−1−<sup>1</sup>sin *x <sup>x</sup>*−0.1*dx*.


**Table 2.** Example 1b: .=1−<sup>1</sup>sin *x* (*<sup>x</sup>*−0.1)<sup>2</sup> *dx*.


## **Example 2.**

$$\mathcal{H}(f,t) = \int\_{-1}^{1} \frac{|\mathbf{x} - \mathbf{0}.5|^{\frac{15}{2}}}{\mathbf{x} - t} d\mathbf{x}, \quad \mathcal{H}^1(f,t) = \oint\_{-1}^{1} \frac{|\mathbf{x} - \mathbf{0}.5|^{\frac{15}{2}}}{(\mathbf{x} - t)^2} d\mathbf{x}, \quad t = 0.3.$$

*In this case, f* ∈ *Z*152 *, and as the results in Tables 3–4 show, the numerical errors agree with the theoretical estimates.*


**Table 3.** Example 2a: .−1−<sup>1</sup> |*<sup>x</sup>*−0.5| 152 *<sup>x</sup>*−0.3 *dx*. Exact value −3.29987610310676.

**Table 4.** Example 2b: .=1−<sup>1</sup>|*<sup>x</sup>*−0.5| 152 (*<sup>x</sup>*−0.3)<sup>2</sup> *dx*.


## **Example 3.**

$$\mathcal{H}(f,t) = \int\_{-1}^{1} \frac{\exp(\mathbf{x})\sin(\mathbf{x})}{1+\mathbf{x}^2} \frac{d\mathbf{x}}{\mathbf{x}-t'} \quad t = -0.7.$$

*Here f* ∈ *C*<sup>∞</sup>*. In this test (see Table 5), we want to show the performance of the quadrature rule when m is fixed and s increases, highlighting how we get an improvement, but it seems till to a certain threshold. This behavior will be the subject of future investigations.*


**Table 5.** Example 3: . −<sup>1</sup> exp(*x*) sin(*x*) *dx x*+0.7 .
