NB. Identities for ^.@* and ^.@% and ^.@^ with complex arguments prodid =: 1 = [: ^ ^.@* - +&^. quotid =: 1 = [: ^ ^.@% - -&^. pwrid =: 1 = [: ^ ^.@^ - (* ^.)~ NB. Remark ^ is periodic with period 0j2p1 periodtest =: 2 : '(v"0 y) = v"0 y+m' 0j2p1 periodtest ^ 2 0j3 4j_5 1 1 1 ]xx =: 2 3j_4 5j6 2 3j_4 5j6 ]yy =: _3 _4j_5 6j7 _3 _4j_5 6j7 xx prodid"0 yy 1 1 1 xx quotid"0 yy 1 1 1 xx pwrid"0 yy 1 1 1
NB. Reciprocals are covered by 1 quotid"0 yy and 0 = ^. 1 --Kip Murray Sent from my iPad > On Dec 17, 2013, at 2:09 AM, "Linda Alvord" <lindaalv...@verizon.net> wrote: > > 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
