Kip, you don't need to define log because   ^.   works for complex argunent. 




>________________________________
> Fra: km <k...@math.uh.edu>
>Til: "programm...@jsoftware.com" <programm...@jsoftware.com> 
>Sendt: 16:16 fredag den 18. januar 2013
>Emne: Re: [Jprogramming] Hermitian from triangular
> 
>Linda, you can define
>
>    log =: 13 : '(^. y) % ^. x'
>
>which I bet is the way dyadic ^. is defined.
>
>   10 log 10^i:2
>_2 _1 0 1 2
>   1j2 log 1j2^i:2
>_2j_1.45289e_16 _1j_7.26445e_17 0 1 2j1.45289e_16
>   1j2 ^. 1j2^i:2
>_2j_1.45289e_16 _1j_7.26445e_17 0 1 2j1.45289e_16
>
>I suppose it would take special code to get rid of the tiny imaginary parts 
>here.
>
>Kip
>
>Sent from my iPad
>
>
>On Jan 18, 2013, at 4:33 AM, "Linda Alvord" <lindaalv...@verizon.net> wrote:
>
>> 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.
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