The unwinding number

Abstract

From the Oxford English Dictionary we find that to unwind can mean "to become free from a convoluted state". Further down we find the quotation "The solution of all knots, and unwinding of all intricacies", from H. Brooke (The Fool of Quality, 1809). While we do not promise that the unwinding number, defined below, will solve all intricacies, we do show that it may help for quite a few problems.Our original interest in this area came from a problem in which an early version of D ERIVE was computing the wrong answer when simplifying sin(sin -1 z ), which should always be just z. For z > 1, D ERIVE was getting - z as the answer. This bug has of course long since been fixed.What was happening was that in order to improve internal efficiency, all the inverse trig functions were represented as arctangents. Consulting an elementary book of tables, one finds the identitysin -1 z = tan -1 ( z /√1 - z 2 ). (1)In the same vein, one finds thatsin(tan -1 w ) = w /√1 + w 2 . (2)Substituting equations (1) and (2) into sin(sin -1 z ) and simplifying, we get z /√1 - z 2 1/√1/1 - z 2 , (3)which D ERIVE quite properly refused to simplify to z, because this is not always equal to z (see [2]).The fix in this case was to replace equation (2) withsin(tan -1 w ) = w √1/1 + w 2 . (4)which differs from the original only on the branch cut. See [7] for more discussion. This change allows the simplification of sin(sin -1 z ) to z. Verifying that this approach worked, and indeed trying to understand the problem to begin with, led us to attempt various definitions of a 'branch function'. This introductory problem turned out to be the tip of an iceberg of problems connected with using the principal branch of multivalued elementary functions.