Similarly, the complementary solutions, $\mathcal{K}_\nu$, are defined as $$\mathcal{K}_\nu \equiv \mathrm{Ker}_\nu (x) + i \mathrm{Kei}_\nu (x)$$ $$= \exp(- \nu \pi i / 2) \cdot K_\nu \left(x \cdot \exp(\pi i / 4)\right)$$ where $K_\nu$ is a modified Bessel function of the second kind.
The relationships between $y$ in the differential equation, and the solutions $\mathcal{B}_\nu$ and $\mathcal{K}_\nu$ are as follows $$y = \mathrm{Ber}_\nu (x) + i \mathrm{Bei}_\nu (x)$$ $$= \mathrm{Ber}_{-\nu} (x) + i \mathrm{Bei}_{-\nu} (x)$$ $$= \mathrm{Ker}_\nu (x) + i \mathrm{Kei}_\nu (x)$$ $$= \mathrm{Ker}_{-\nu} (x) + i \mathrm{Kei}_{-\nu} (x)$$
In the case where $\nu=0$, the differential equation reduces to $$x^2 \ddot{y} + x \dot{y} - i x^2y = 0$$ which has the set of solutions: $$J_0 \left(i \sqrt{i} \cdot x\right)$$ $$= J_0 \left(\sqrt{2} \cdot (i-1) \cdot x / 2\right)$$ $$= \mathrm{Ber}_0 (x) + i \mathrm{Bei}_0 (x) \equiv \mathcal{B}_0$$
This package has functions to calculate $\mathcal{B}_\nu$ and $\mathcal{K}_\nu$.
Keir, Beir