Take a partition of $\Bbb R^2_{\gt 0}$ by the union of functions indexed by real $t\ge 0$
$$\mathcal F:=\bigg \lbrace \mathcal M[\chi_t(x)]\cup \mathcal M\bigg[\frac{1}{1-\chi_t(x)}\bigg] \bigg \rbrace$$
where
$$\mathcal M[\chi_t(x)]:=\int_{(0,1)} \chi_t(x)x^{s-1}~dx= 2\sqrt{\frac{t}{s}}K_1(2\sqrt{ts})=\Phi_t(s)$$
and
$$\mathcal M\bigg[\frac{1}{1-\chi_t(x)}\bigg] :=\int_{(0,1)} \frac{1}{1-\chi_t(x)}x^{s-1}~dx= \Psi_t(s)= \sum_{n=0}^\infty \Phi_{tn}(s)=\sum_{n=0}^\infty 2\sqrt{\frac{tn}{s}}K_1(2\sqrt{ts})$$
for $K_1$ Bessel function.
Consider the functions indexed again by $t$
$$H_t(x)=-\frac{\Phi'{_t}(s)}{\Phi{_t}(s)\Psi_t(s)} $$
This set of functions forms a real analytic partition (even a foliation) of $\Bbb R^2_{\gt 0}$ we have that $$\bigcup_{t\ge 0} H_t(x)=\Bbb R^2_{\gt 0}$$
But I am curious, what is the reason for the oscillations in the plot? Is this an issue with Desmos?
Here is a picture of the partition. Link to the plot: Desmos Plot.
