Numerical accuracy

To calculate residues, we take circular contour integrals of radius $r_0$:

$$z=r_0e^{i\phi}.$$ (53)

The integrals are evaluated using the Fomenko discretization method [42,43], whereby values of integrands are summed up at gauge angles $\phi_k=\frac{k\pi}{L}$ for $k=0,\ldots,L-1$. This corresponds to the upper half circle in the complex $z$-plane and, as discussed in Sec. 3.2, only the real part of the integral is kept. For analytic integrands, the Fomenko method delivers exact results up to admixtures of wave functions with $N\pm{L},N\pm{2L},N\pm{3L},\ldots$ particles. The main question in applying this method to non-analytic integrands, which have poles in the complex plane, is to what extend can it deliver equally accurate numerical results.

The Fomenko method clearly fails when there is a pole (28) lying just on the integration contour, $r_0=\vert z_n\vert$, and an even number of points $L$ is used. In such a case, the integration point with $k$=$L/2$ is located exactly at the pole of the integrand. Therefore, in most practical calculations, an odd number of integration points, most often $L=7$ or 9, was used.

However, a more stringent condition on $L$ results from the fact that the discretization method must fail whenever the integrand varies too rapidly between two neighboring integration points. Therefore, the spacing between points ${\pi r_0}/{L}$ must be appropriately smaller than the distance from the pole. For odd values of $L$, the integration points corresponding to $k=(L\pm1)/2$ are closest to the imaginary axis; hence, one arrives at the condition

$$\frac{\pi r_0}{L} < \sqrt{(r_0-\vert z_n\vert)^2+\left(\frac{\pi r_0}{2L}\right)^2} ,$$ (54)

or

$${L} > \frac{\sqrt{3}\pi r_0}{\sqrt{4}\vert r_0-\vert z_n\vert\vert} .$$ (55)

In the present study, a large number of $L=93$ integration points was used, which allows for calculating the contour integrals with radii $r_0$ that differ by as little as 3% from the position of the closest pole $\vert z_n\vert$.