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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