Linda, base x logarithm is 4 : ' (^.y)%(^.x)' for nonzero complex arguments x and y , except when x=1. (^.1 is 0, and thou shallt not divide by zero).
_10^.5j6 0.493362j_0.292664 _10^0.493362j_0.292664 5.00001j6.00002 >________________________________ > Fra: Linda Alvord <lindaalv...@verizon.net> >Til: programm...@jsoftware.com >Sendt: 11:33 fredag den 18. januar 2013 >Emne: Re: [Jprogramming] Hermitian from triangular > >Kip, I just got back to a different and interesting sidetrack on this long >thread. What a simple way to write a proof in J. > > _1 = ^ 0j1 * o. 1 >1 > > (0j1 * o.1) = ^. _1 >1 > > >Therefore: Negative numbers can have logarithms to the base e > >Can they also have common logs? > >Also, It makes you wonder if there isn't some sequence out there somewhere >where there is an ordered sequence of complex numbers: > > i:2 >_2 _1 0 1 2 > > i:0j2 > >Happy wandering and pondering. > >Linda > > >-----Original Message----- >From: programming-boun...@forums.jsoftware.com >[mailto:programming-boun...@forums.jsoftware.com] On Behalf Of km >Sent: Thursday, January 17, 2013 9:31 AM >To: programm...@jsoftware.com >Subject: Re: [Jprogramming] Hermitian from triangular > >Linda, about logarithms of negative numbers > >First of all, you know the number e =: ^ 1 and you know ^ y is e^y . You >may not know that ^ x j. y by definition is (^ x) * (cos + 0j1 * sin) y >where cos =: 2&o. and sin =: 1&o. . I first learned this in a college >math class called Complex Analysis. A good reference is E. B. Saff and A. >D. Snider, Fundamentals of Complex Analysis, Pearson Education, Inc. 2003. > >Anyway, a famous identity in higher math is > > _1 = ^ 0j1 * o. 1 >1 > >which should tell you that > > (0j1 * o.1) = ^. _1 >1 > >i.e., negative numbers can have logarithms to the base e . For more on >this, please see Saff and Snider's Chapter 3. > >Kip Murray > >Sent from my iPad > > >On Jan 17, 2013, at 4:22 AM, "Linda Alvord" <lindaalv...@verizon.net> wrote: > >> Isn't the log of negative numbers indefined? >> >> This is a problem: >> >> %1&o.+0 >> _ >> %1&o.-0 >> _ >> >> This is nice! >> >> %1&o.%_ >> _ >> %1&o.%__ >> __ >> >> >> The csc is very small for negative numbers close to zero and very >> large for very small positive numbers. >> >> Linda >> >> -----Original Message----- >> From: programming-boun...@forums.jsoftware.com >> [mailto:programming-boun...@forums.jsoftware.com] On Behalf Of Bo >> Jacoby >> Sennt: Thursday, January 17, 2013 3:37 AM >> To: programm...@jsoftware.com >> Subject: Re: [Jprogramming] Hermitian from triangular >> >> Henry, How is negative zero different from positive zero when taking >> the log? >> ^.%__ NB. log -0 >> __ >> ^.%_ NB. log +0 >> __ >> >> >> - Bo >> >> >>> ________________________________ >>> Fra: Henry Rich <henryhr...@nc.rr.com> >>> Til: programm...@jsoftware.com >>> Sendt: 0:38 torsdag den 17. januar 2013 >>> Emne: Re: [Jprogramming] Hermitian from triangular >>> >>> Negative zero makes sense as a last vestige of gradual underflow; and >> anyway, it's well-behaved: it looks like 0 except when you take the >> log, reciprocal, or square root. In any normal computation, it goes >> away. In contrast, NaN messes up anything it touches. >>> >>> I think we've had negative 0 in J forever. If NaN is a data virus, >>> -0 is a >> virus that has been inserted into our DNA. >>> >>> Henry Rich >>> >>> On 1/16/2013 4:45 PM, Raul Miller wrote: >>>> On Wed, Jan 16, 2013 at 4:35 PM, Henry Rich <henryhr...@nc.rr.com> >wrote: >>>>> Negative zero isn't a bug, it's a feature that numerical types, >>>>> especially William Kahan, wanted to get into IEEE-754 to help out >>>>> some things. I'm not expert enough to explain. >>>> >>>> Something similar could be said about NaN. >>> --------------------------------------------------------------------- >>> - 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
