Dan Bron wrote: > PS: > > I'm not a mathematician, but it seems to me that the right solution is to > express ^._1 as 0 j. (+,-) 1p1 . But no one does. At least not in > front of me.
The problem here is that a function f has an inverse function only if f is 1-1. It always has an inverse relation. If f is not 1-1, and we want an inverse function, we restrict its domain to give a new function g hich is 1-1, and then take the inverse of that. The simplest example is f(x)=x^2, which is not 1-1 (e.g. f(-1)=f(1)=1). We restrict its domain to give the new function g(x)=x^2, x>:0, and call the inverse of g the square root. The choice of restriction is somehat arbitrary, but there are established conventions for trigonometric functions, giving arcsin, arccos, etc. It's a bit more complicated for ^ and ^. For real values, ^ is 1-1 on the nose, so no restriction is needed to define ^. For y real, ^ j. y is defined by Euler's formula as (cos y)+j.(sin y). This has period 2p1, so to make it 1-1 you cannot include 0j1p1 and 0j_1p1 in the domain of the restriction. This is why ^. _1 is never expressed as as 0 j. (+,-) 1p1 . Complex log defined via a specific restriction is not really so useful a function. What is important is that complex exponential has a local inverse. Eldon's point, and my code on arctan _1r1, were concerned with variant defintions of arctan. Arctan is normally defined by restricting the domain of tangent to the open interval (-1r2p1, 1r2p1), so it always gives an angle in the first or fourth quadrant. In many applications, you want the angle from the x-axis to the point (x,y), which can be in any quadrant. This is basically arctan y%x, except it takes sign into account. Library functions called atan2 usually do this, and J does it via angle (12&o.). Then arctan _1 is _1r4p1 (in the fourth quadrant), same as the angle to (1,_1), but different from the angle to (_1,1). Best wishes, John ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
