This study aims to investigate the morphism and stability of a family of new transcendental functions, namely H functions, which have been proven a suitable vehicle for expressing the exact solutions of drug concentration over time of a one-compartment pharmacokinetic (PK) model with sigmoidal Hill elimination. Restricting in the real values of Hill coefficients \(\alpha \) for the H functions, the real branches of the H functions are identified and their stabilities are analyzed. The number, shape, monotonicity, and concavity of all real branches are determined, as well as the stability of each branch when \(\alpha \) is changed. The results show that the principal real branch \(H_0(s)\) is the unique stable branch for \(\alpha \in \mathbb {R}\) , while other real branches can only exist for \(\alpha \in \mathbb {Q}\) and classified therein. A numerical experiment is conducted between the proposed H function and other commonly used solvers, including differential equation solvers (ode45), and the results indicate that the former is more reliable and has an acceptable level of induced error.
