On the best rational approximation of the function $z^{-1/2}$ on the union of the two segments of the real line

When constructing absorbing boundary conditions for solving discretized hyperbolic PDEs, one needs to know good $[n-1/n]$-type rational approximants to the function $z^{-1/2}$ defined with the slit along the negative imaginary semiaxis. The compact set $K$, on which the chosen branch of the function $z^{-1/2}$ is approximated, equals the union of two real line segments separated by the imaginary axis: $K=[a_1,b_1]\cup[a_2,b_2]$, $a_1<b_1<0<a_2<b_2$. An upper (constructive) and lower approximation error bounds will be demonstrated; they have been in a good agreement with each other. Relative rational approximation problems will also be considered.