[Jprogramming] A complex question?

[km]

Off topic: I now know that taking the logarithm of the reciprocal of a non-zero real number changes the sign of the real part of the logarithm of the original number:

   csrp NB. change sign of real part
(1r2 * -@(+ +) + (- +))"0
   csrp 1j2 _3j_4
_1j2 3j_4
   ]rr =: 1 % 4 2 1 0.5 0.25  NB. non-zero reals
0.25 0.5 1 2 4
   (^.@% -: csrp@^.) rr
1
   (^.@% -: csrp@^.) -rr
1

--Kip

Sent from my iPad

> On Dec 16, 2013, at 7:02 PM, "Dan Bron" <j at bron.us> wrote:
> 
> Not sure.  I suppose instead of
> 
>    -@^.@(+/&.:*:)
> 
> we could write:
> 
>    ^.@%@(+/&.:*:)
> 
> or even:
> 
>    ^.@(+/&.:(*: :. (^&_0.5) ) )
> 
> But I'm not sure what this buys us.  
> 
> -----Original Message-----
> From: programming-bounces at forums.jsoftware.com
> [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of km
> Sent: Monday, December 16, 2013 7:49 PM
> To: programming at jsoftware.com
> Subject: Re: [Jprogramming] A complex question?
> 
> Dan, I haven't been following this thread, but know that minus the logarithm
> of a positive number is the logarithm of the reciprocal.  Is that relevant?
> 
>   ^. 1r4 1r2 1 2 4
> _1.38629 _0.693147 0 0.693147 1.38629
> 
> 
> --Kip
> 
> Sent from my iPad
> 
>> On Dec 16, 2013, at 3:42 PM, Dan Bron <j at bron.us> wrote:
>> 
>> Raul wrote:
>>> Is there a better way of doing this?
>>>  {: +. r.inv j./1 1
>> 
>> Marshall responded:
>>> You can also use (+/&.:*:) in place of |@j./ , leaving you with 
>>> -@^.@(+/&.:*:)"1
>> 
>> Raul wrote:
>>> Experimenting: the - is necessary and the ^. is not necessary. 
>>> (I do not get a hexagon without the minus, I do get a hexagon without 
>>> the ^.).
>> 
>>> Immediately after writing this I realized the - is also unnecessary - 
>>> changing >./ to <./
>> 
>> What I love is that through some simple trig and a few experiments, we 
>> got from {:@+.@(r.^:_1)@(j./) to +/&.:*: .
>> 
>> I suppose I find this particularly gratifying because I spent some 
>> time trying to restate Raul's phrase in terms of simple arithmetic 
>> operations, staying entirely in the real domain, and I eventually 
>> reproduced Marshall's verb.  Having spent so much time "simplifying", 
>> when I got the final, irreducible result, I wondered at the need for 
>> -@^. , and what its physical interpretation was.
>> 
>> Raul's original verb could be rendered in English as "the length 
>> component of a polar coordinate (initially specified in Cartesian 
>> terms)".  Why should that length be expressed as the negative log of a 
>> distance?  Why not, as Don put it, "the raw distance"?
>> 
>> I know there are subtle and beautiful connections between the 
>> trigonometric and exponential functions, and the e hidden in r. is one 
>> expression of that.  But I'm still not seeing the fundamental physical
> interpretation.
>> In other words, I wasn't surprised with the -@^. disappeared in Raul's 
>> use case; I might've been more surprised if it'd persisted.
>> 
>> Anyone want to help me see it? Maybe the best illustration would be a 
>> concrete use case where the -@^. isn't superfluous - one where where 
>> it is not only necessary, but inevitable?
>> 
>> That is, a use case where -@^. has obvious physical interpretation, 
>> when applied to the distance. Ideally one like Raul's, which 
>> ultimately didn't involve complex numbers (i.e. a real-valued binary 
>> [dyadic] operation on real numbers).
>> 
>> -Dan
>> 
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