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" <[email protected]> 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: [email protected]
> [mailto:[email protected]] On Behalf Of km
> Sent: Thursday, January 17, 2013 9:31 AM
> To: [email protected]
> 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" <[email protected]> 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: [email protected]
>> [mailto:[email protected]] On Behalf Of Bo
>> Jacoby
>> Sennt: Thursday, January 17, 2013 3:37 AM
>> To: [email protected]
>> 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 <[email protected]>
>>> Til: [email protected]
>>> 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 <[email protected]>
> 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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