Henry Rich wrote: > The interpreter thinks > > 11&o. b. _1 > _11&o. > > but it ain't so. Circle functions 9-12 have no inverse that > I can see. Circle functions _9-_12 do, but they're not > the functions 9-12&o. that the interpreter uses. >
The following explanation at the bottom of the Dictionary page for o. appears flawed: "The principal domain of the inverse of a non-monotonic function is restricted. The limits on real domains of arcsine and arccos may be obtained as follows..." If a function is monotonic, then it is 1-1, but it is the latter condition that is necessary for it having an inverse. Monotonicity makes no sense if the domain is the complex numbers, as with 9&.o. . The usual terminology is to take a function f and restrict its domain and codomain to D and C so that the new function f* is 1-1 and onto, and thus has an inverse. D is called the principal domain of f* and C the (set of) principal values. The inverse of f* has domain C and codomain D and is what we loosely call "the inverse of f". In the second sentence quoted, it is not the limits on real domains of arcsin and arccos that are being described, but the limits on the real domains of sin and cos. I would suggest the following rewording: "A function which is not 1-1 is restricted to a principal domain. The limits on principal values of arcsine and arccos may be obtained as follows..." Since conventions differ as to what principal domains are, even with quite standard functions, it might be worth spelling them out, as is done for sin and cos, the easiest ones to guess. This is even more worthwhile for less obvious cases like 9&o. . Best wishes, John ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
