Principal Branches of Inverse Trigonometric and Inverse Hyperbolic Functions

TL;DR

This work develops explicit principal branches for three square-root functions and constructs principal branches for the inverse trigonometric and inverse hyperbolic functions using complex-analytic continuation within DLMF conventions. It provides closed-form, axis- and cut-aware expressions for $\sqrt{z-1}$, $\sqrt{z+1}$, $\sqrt{z^2-1}$, and $\sqrt{1-z^2}$, and then derives their corresponding inverse functions with derivatives and antiderivatives, linking them through a unifying framework of associated functions $X$, $Y$, and $Z$. The paper also details the behavior on branch cuts, reflection formulas, and special values on the real and imaginary axes, including multiple two-valued regimes on the cuts. By systematizing these principal branches, the work broadens the DLMF's scope and provides tools to reduce the complexity of complex-variable calculations in applied and theoretical contexts.

Abstract

We develop principal branches for three key square root functions and for the inverse trigonometric and inverse hyperbolic functions. The three square root branches are integral to defining the inverse function branches, their derivatives, and their antiderivatives. Complex analysis is used to turn the definitions of the principal branches into concrete expressions. We take the standard reference in this area to be the NIST Digital Library of Mathematical Functions (DLMF). We adopt the notation for, and the definitions of, the principal branches of the inverse functions in the DLMF. Our goal is to widen the scope of the results in the DLMF while at the same time lowering the complex variables burden on the average DLMF user.