Look at ^. _2 _0.5 0.693147j3.14159 _0.693147j3.14159
^ +/ ^. _2 _0.5 NB. "Practically 1" on my iPad 1j_2.44929e_16 ^ 0 0j2p1 NB. Second result is "practically 1" 1 1j_2.44929e_16 Because verb ^ has period 0j2p1 a number can have many complex logarithms. Verb ^. picks out the one whose imaginary part imp satisfies _1p1 < imp and imp <: 1p1 . For a straightforward treatment of complex logarithms see Section 2.6 beginning page 46 of Polya and Latta's 1974 book Complex Variables. --Kip Sent from my iPad >> On 17/12/2013 12:09 AM, Linda Alvord 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
