Principal Branches of Inverse Trigonometric and Inverse Hyperbolic Functions

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.