Multivalued Inverse Functions

Abstract

Most mathematics teachers have experienced the difficulties of teaching the inverse trigonometric functions to students. The same difficulties that students experience have been experienced by computer algebra systems. Recent studies of the Lambert W function have suggested a modification of the usual notation that makes multivalued functions easier to work with and to teach. There are immediate benefits when teaching the inverses of sine, cosine, and tangent - all functions having real periods. Functions with complex periods can be treated using the same approach. The main idea is to label each branch of the inverse function. For the Lambert W function, this step is unavoidable; for the elementary functions it is avoidable - and it has been until now - but labelling does make the subject easier to understand. Thus the infinite number of solutions of the equation sin y=x are called y=invsin(x,k) where the traditional arcsin name is not used to avoid confusion. The different solutions are obtained by giving k different values. For example, the equation sin(y) =1/2 has solutions y=invsin(1/2,0)=pi/6, y=invsin(1/2,1)=5pi/6 and y=invsin(1/2,2)=13pi/6. In this scheme, the questions that always cause the most difficulty to students and computer systems, namely, does arcsin mean all solutions or only one, and does arcsin(sin(x))=x, no longer arise.