We have chosen a special ray $(\eta_{1},\eta_{2})=(\sqrt{-1}b,\sqrt{-1}b)$ to calculate the asymptotic value of $\beta_{p3}+\beta_{p4}$. More generally $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ divide the plane $\mathbb{R}^{2}_{y_{1},y_{2}}$ into three sectors, and the asymptotic value of function

$$\beta_{p3}+\beta_{p4}=\frac{-2\pi ie^{2\pi i\eta_{p}}}{1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}}+K_{p}(a)$$

along the ray $\{(y_{1},y_{2})=s\vec{e}\text{ for }s>0\}$ specified by a directional vector $\vec{e}$ depends on which sector $\vec{e}$ belongs to, and can have a jumping discontinuity as we cross $\mathfrak{D}_{i}$. This is known as Stokes phenomenon in complex analysis.