So now I realize I don't understand how to get the reciprocal of a complex number.
% _3j4 _0.12j_0.16 Linda -----Original Message----- From: programming-boun...@forums.jsoftware.com [mailto:programming-boun...@forums.jsoftware.com] On Behalf Of Linda Alvord Sent: Tuesday, December 17, 2013 3:09 AM To: programm...@jsoftware.com Subject: Re: [Jprogramming] A complex question? Is this a true statement? NB. Taking the logarithm of the reciprocal of a NB. number changes the sign of all parts of the logarithm NB. the original number. N=: 4 2 1 0 0.5 0.25 ^. N 1.38629 0.693147 0 __ _0.693147 _1.38629 ]R=: % N 0.25 0.5 1 _ 2 4 ^. R _1.38629 _0.693147 0 _ 0.693147 1.38629 C=:1j1 _1j1 _1j_1 1j_1 ^.C 0.346574j0.785398 0.346574j2.35619 0.346574j_2.35619 0.346574j_0.785398 ]RC=: % C 0.5j_0.5 _0.5j_0.5 _0.5j0.5 0.5j0.5 ^. RC _0.346574j_0.785398 _0.346574j_2.35619 _0.346574j2.35619 _0.346574j0.785398 Linda From: programming-boun...@forums.jsoftware.com [mailto:programming-boun...@forums.jsoftware.com] On Behalf Of km Sent: Monday, December 16, 2013 10:54 PM To: programm...@jsoftware.com Subject: Re: [Jprogramming] A complex question? Off topic: I now know that taking the logarithm of the reciprocal of a non-zero real number changes the sign of the real part of the logarithm of the original number: csrp NB. change sign of real part (1r2 * -@(+ +) + (- +))"0 csrp 1j2 _3j_4 _1j2 3j_4 ]rr =: 1 % 4 2 1 0.5 0.25 NB. non-zero reals 0.25 0.5 1 2 4 (^.@% -: csrp@^.) rr 1 (^.@% -: csrp@^.) -rr 1 --Kip Sent from my iPad > On Dec 16, 2013, at 7:02 PM, "Dan Bron" <j...@bron.us> wrote: > > Not sure. I suppose instead of > > -@^.@(+/&.:*:) > > we could write: > > ^.@%@(+/&.:*:) > > or even: > > ^.@(+/&.:(*: :. (^&_0.5) ) ) > > But I'm not sure what this buys us. > > -----Original Message----- > From: programming-boun...@forums.jsoftware.com > [mailto:programming-boun...@forums.jsoftware.com] On Behalf Of km > Sent: Monday, December 16, 2013 7:49 PM > To: programm...@jsoftware.com > Subject: Re: [Jprogramming] A complex question? > > Dan, I haven't been following this thread, but know that minus the > logarithm of a positive number is the logarithm of the reciprocal. Is that relevant? > > ^. 1r4 1r2 1 2 4 > _1.38629 _0.693147 0 0.693147 1.38629 > > > --Kip > > Sent from my iPad > >> On Dec 16, 2013, at 3:42 PM, Dan Bron <j...@bron.us> wrote: >> >> Raul wrote: >>> Is there a better way of doing this? >>> {: +. r.inv j./1 1 >> >> Marshall responded: >>> You can also use (+/&.:*:) in place of |@j./ , leaving you with >>> -@^.@(+/&.:*:)"1 >> >> Raul wrote: >>> Experimenting: the - is necessary and the ^. is not necessary. >>> (I do not get a hexagon without the minus, I do get a hexagon >>> without the ^.). >> >>> Immediately after writing this I realized the - is also unnecessary >>> - changing >./ to <./ >> >> What I love is that through some simple trig and a few experiments, >> we got from {:@+.@(r.^:_1)@(j./) to +/&.:*: . >> >> I suppose I find this particularly gratifying because I spent some >> time trying to restate Raul's phrase in terms of simple arithmetic >> operations, staying entirely in the real domain, and I eventually >> reproduced Marshall's verb. Having spent so much time "simplifying", >> when I got the final, irreducible result, I wondered at the need for >> -@^. , and what its physical interpretation was. >> >> Raul's original verb could be rendered in English as "the length >> component of a polar coordinate (initially specified in Cartesian >> terms)". Why should that length be expressed as the negative log of >> a distance? Why not, as Don put it, "the raw distance"? >> >> I know there are subtle and beautiful connections between the >> trigonometric and exponential functions, and the e hidden in r. is >> one expression of that. But I'm still not seeing the fundamental >> physical > interpretation. >> In other words, I wasn't surprised with the -@^. disappeared in >> Raul's use case; I might've been more surprised if it'd persisted. >> >> Anyone want to help me see it? Maybe the best illustration would be a >> concrete use case where the -@^. isn't superfluous - one where where >> it is not only necessary, but inevitable? >> >> That is, a use case where -@^. has obvious physical interpretation, >> when applied to the distance. Ideally one like Raul's, which >> ultimately didn't involve complex numbers (i.e. a real-valued binary >> [dyadic] operation on real numbers). >> >> -Dan >> >> --------------------------------------------------------------------- >> - For information about J forums see >> http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
