> 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..."
This is not correct either. For example, 11 o. y , which computes the imaginary part of y, is not 1-1 but is not restricted to a principal domain. (11 o. y works for any complex number y.) What is restricted is that the inverse is an inverse only for a significant subdomain. So _11 o. y is the inverse of 11 o. y for the set of imaginary numbers, complex numbers with real part 0. See the dictionary page for / for a related discussion on neutrals (identity elements). ----- Original Message ----- From: John Randall <[EMAIL PROTECTED]> Date: Friday, June 30, 2006 5:02 am Subject: Re: [Jbeta] _11&o. is not the inverse of 11&o. > 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. . ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
