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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> .
>
>
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