> You may not know that ^ x j. y by definition is (^ x) * (cos + 0j1 * sin) y
This is too complex a definition for my taste. I prefer one where (for example) ^z is defined to be a function which is equal to its derivative, and derive the equation you cited as a theorem. And ^. is the inverse of ^ . Regarding 0 = 1 + ^ 1p1 * 0j1, see http://www.jsoftware.com/jwiki/Essays/Euler's_Identity In J7.01, you can do this: ^@o. j. 0.5 * i. 3 4 1 0j1 _1 0j_1 1 0j1 _1 0j_1 1 0j1 _1 0j_1 ^@o. j. 2e9 + 0.5 * i. 3 4 1 0j1 _1 0j_1 1 0j1 _1 0j_1 1 0j1 _1 0j_1 On Thu, Jan 17, 2013 at 6:30 AM, km <k...@math.uh.edu> wrote: > 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
