D. Character of the singularity on the integration path

In this section we determine the leading contribution to the logarithmic (and thus integrable) singularity on the integration path that appears in the evaluation of the integral in (24). The logarithmic character stems from the definition of the inverse hyperbolic tangent in terms of the logarithm and from the fact that its argument is a rational function with the following properties⁸:

$$\lim _{x \to \vert \Theta \vert} {\mathcal{F} (x ; \Theta)... ...4 \Theta ^{2}}}{ x ^{3} - 2 x + 2 \vert \Theta \vert} = 1 ,$$

i.e.

$${\mathcal{F}} ( \vert \Theta \vert ; \Theta ) = 1$$ (37)

To extract the behaviour of the above function around the point $x = \vert \Theta \vert$ we develop it in power series. Let us set

$${\mathcal{R}} (x ; \Theta) = x ^{6} - 4 \vert \Theta \vert x ^{3} + 4 \Theta ^{2} x ^{2} \quad \mathrm{with} \quad {\mathcal{R}} (\vert \Theta \vert ; \Theta) = \Theta ^{6}$$ (38)

$${\mathcal{D}} (x ; \Theta) = x ^{3} - 2 x + 2 \vert \Theta \vert \quad \mathrm{with} \quad {\mathcal{D}} (\vert \Theta \vert ; \Theta) = \vert \Theta \vert ^{3} .$$ (39)

Then it follows

$${\mathcal{R}} ' (x ; \Theta) = 6 x ^{5} - 12 \vert \Theta \vert x ^{2} + 8 \Theta ^{2} x \quad \... ... \Theta \vert ; \Theta) = 6 \vert \Theta \vert ^{5} - 4 \vert \Theta \vert ^{3}$$ (40)

$${\mathcal{D}} ' (x ; \Theta) = 3 x ^{2} - 2 \quad \mathrm{so\ that} \quad {\mathcal{D}} ' (\vert \Theta \vert ; \Theta) = 3 \Theta ^{2} - 2 .$$ (41)

and

$${\mathcal{R}} '' (x ; \Theta) = 30 x ^{4} - 24 \vert \Theta \vert x + 8 \Theta ^{2} \quad \mathrm... ...\mathcal{R}} '' (\vert \Theta \vert ; \Theta) = 30 \Theta ^{4} - 16 \Theta ^{2}$$ (42)

$${\mathcal{D}} '' (x ; \Theta) = 6 x \quad \mathrm{so\ that} \quad {\mathcal{D}} '' (\vert \Theta \vert ; \Theta) = 6 \vert \Theta \vert .$$ (43)

We then compute ${\mathcal{F}} ' (\vert \Theta \vert ; \Theta)$ and ${\mathcal{F}} '' (\vert \Theta \vert ; \Theta)$, i.e. the first and second derivatives of the argument of the inverse hyperbolic tangent. Since

$${\mathcal{F}} ' = \frac{{\mathcal{R}} ' {\mathcal{D}} - 2... ... {\mathcal{D}} '} {2 {\mathcal{R}} ^{1/2} {\mathcal{D}} ^{2}}$$

we obtain

$${\mathcal{F}} ' ( \vert \Theta \vert ; \Theta ) = 0 .$$ (44)

Moreover

$${\mathcal{F}} '' = \frac{ 2 \left( {\mathcal{R}} '' {\... ...D}} ' \right) } {4 {\mathcal{R}} ^{3/2} {\mathcal{D}} ^{3}}$$

and we get

$${\mathcal{F}} '' ( \vert \Theta \vert ; \Theta) = - \frac{4}{\Theta ^{6}} \left( 1 - \Theta ^{2} \right) .$$ (45)

The expansion of ${\mathcal{F}}$ around $x = \vert \Theta \vert$ up to second order can then be written, using (38), (45) and (46), as

$${\mathcal{F}} (x ; \Theta) = 1 - \frac{2 \left( 1- \The... ...} + {\mathcal{O}} \left( x - \vert \Theta \vert \right) ^{3}$$

and inserting this result inside the expression of the hyperbolic tangent in terms of logarithms, we can easily see, as expected, that the singularity is integrable.