Re: Possible Bug in ArcTan ?

*To*: mathgroup at smc.vnet.net*Subject*: [mg64936] Re: Possible Bug in ArcTan ?*From*: "David W. Cantrell" <DWCantrell at sigmaxi.org>*Date*: Wed, 8 Mar 2006 00:59:56 -0500 (EST)*References*: <du6o44$5rg$1@smc.vnet.net> <du83m5$sv3$1@smc.vnet.net> <du8are$fp7$1@smc.vnet.net> <dubgv0$fm7$1@smc.vnet.net> <due86a$9vj$1@smc.vnet.net> <dujr11$8ou$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Paul Abbott <paul at physics.uwa.edu.au> wrote: > In article <due86a$9vj$1 at smc.vnet.net>, > "Jens-Peer Kuska" <kuska at informatik.uni-leipzig.de> wrote: > > > why you want avoid the two-argument form. The two-argument form help > > a lot during programing, because one has not to type > > > > If[x=!=0,ArcTan[y/x]] > > > > and a division by zero is in the most programing languages > > a very hard and evil error. > > There is no "division by zero" error using the form > > 2 ArcTan[y/(x+Sqrt[x^2+y^2])] > > except when x == y == 0 -- where the result is undefined anyway. First, there will be division by zero if y is zero and x is nonpositive. See below. Second, the very special case when both x and y are zero is interesting. Of course, one could well argue that we should _want_ the result to be undefined, that is, Indeterminate in Mathematica. And indeed, that's what Paul's expression gives. But there's also a good argument that we should want ArcTan[x,y] to be defined even when both x and y are zero. That's what Mathematica does: ArcTan[0,0] yields Interval[{-Pi,Pi}]. IMO, the only result which would be slightly better would be the "half-open, half-closed" interval (-Pi, Pi], which I suppose is unavailable in Mathematica. > However, as David Cantrell points out, this form (only) fails if y is > zero and x is negative (returning 0 instead of Pi), and so must be > considered separately. Actually, when x is negative and y is zero, we also have division by zero. But then we're dividing into zero as well, and so in Mathematica the result is neither Pi nor 0, but rather Indeterminate. David