On 01/26/2010 03:21 PM, Sheldon Newhouse wrote:
> On 01/26/2010 02:24 PM, Barton Willis wrote:
>
>> If domain(log) = (0,inf), then limit(log(x),x,0) = -inf is
>> correct. This is agrees with the limit definition in T. Apostal, for
>> example. And it agrees, I think with
>>
>> http://functions.wolfram.com/ElementaryFunctions/Log/03/02/
>>
>> Now if domain(log) = complex - {0}, the correct value is complex
>> infinity (infinity in Maxima); see the section "Complex
>> analysis" in http://en.wikipedia.org/wiki/Infinity
>>
>> Of course, largely the problem is that limits are defined for
>> functions, not their formulae. We have limit(n in Z |--> sin(n),
>> infinity) = 0, but limit(n in R |--> sin(n), infinity) is undefined
>> (here Z = set of integers and R = set of reals).
>>
>> Barton
>>
>> maxima-bounces at math.utexas.edu wrote on 01/26/2010 12:49:28 PM:
>>
>>
>>
>>> [image removed]
>>>
>>> [Maxima] limit(log(x),x,0)?
>>>
>>> Raymond Toy
>>>
>>> to:
>>>
>>> Maxima List
>>>
>>> 01/26/2010 12:49 PM
>>>
>>> Sent by:
>>>
>>> maxima-bounces at math.utexas.edu
>>>
>>> What is limit(log(x),x,0)? Maxima currently returns infinity. I
>>> suppose that's correct, but with a default domain of real, shouldn't
>>> this be undefined?
>>>
>>> Ray
>>>
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>>>
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>>
>>
>>
> OK, let's discuss this a bit. For details, see Ahlfors (Complex
> Analysis) or some similar book.
>
> In the complex case, (as in the real case) the log should be the
> inverse function for z -> exp(z).
>
> In the real case x -> exp(x) is strictly increasing, so there is no
> problem defining the inverse function on R (the set of real numbers).
>
> However, in the complex case, the function z -> exp(z) is far from
> one-to-one. In fact, it assumes every finite non-zero complex value
> infinitely often.
>
> The easiest way to visualize this is to write a non-zero complex
> number w in polar form as
> w = r*exp(%i * theta) where r = abs(z) and theta is some real number
> (thought of as an angle) called an 'argument' of w.
> Since r> 0, and r is real, we can write r = exp(r1) for some real
> number r1.
>
> Then, w = exp(r1)*exp(%i*theta) = exp(r1+%i*theta).
>
> For any theta_1 = theta + n*2*%pi, with n some integer, we get that
> w = exp(r1 + %i*theta_1) = exp(r1 + %i*theta).
>
>
>
> Let C be the set of complex numbers.
>
> So, to define the complex log function, we need to specify domains D1,
> D2 in C such that f(z) = exp(z) maps D1 bijectively onto D2 and then
> define
> (what is usually called a "branch" of log(z)) to be the inverse map
> from D2 onto D1.
>
>
>
> Typically, one chooses D2 to be the complement of an infinite half-line
> joining 0 to infinity in C and one obtains D1 as a strip of the form
> { (x,y): x \in R, theta_1< y< theta_1 + 2*%pi } for some fixed
> real number theta_1
>
> In the case where theta_1 = -%pi, one speaks of the "principal branch"
> of log(z), and writes this Log(z).
>
> HTH,
> -sen
>
>
>
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>
>
Whoops! The inverse of the real exp function of course has to be
defined on the image of exp (i.e., the positive reals) not on the set of
real numbers as stated above.
Email and the internet make it too easy to think, type, and not check
before you send.
-sen