[TI-M] Re: Logs of negative values

[Prev][Next][Index][Thread]

[TI-M] Re: Logs of negative values


Well, it is a matter of what you want to define the function on.  If you
are talking about f : R -> R (the function f mapping from reals to
reals)... Then log(-R) is undefined in terms of a real valued function.
If you want to talk about the complex valued logarithm, you have a lot
more values to play with, and in fact you can represent the result.

I suppose that this is analogous to roots of functions.  x^2+1 = y has no
real roots, but it has two complex roots, namely i and -i.

Now, back to your example.  You know that Log is the inverse of the
exponential.  Let's check the value you got.  One fact about complex
exponentials is that the identity (Euler's) is that

e^(i x ) = cos(x) + i sin(x)

Taking e^(i pi) = cos(pi) + i sin (pi) = -1
So indeed, it is true.


The identity shown above can we derived from the polar form of the complex
number.  Any complex number, because of its relation to a 2d vector space
can be expressed

z = r e^(i x)

We do have z = x + iy (a complex number).  But we can do
z = r (cos x + i sin x)
(x is like the angle)
But taking x above as x =0 and y = t we get
e^(i t) = cos t + i sin t...



-- 
Andy Selle <aselle@ticalc.org>
   Programming and System Administration, Survey Editor, Accounts Manager
   the ticalc.org project - http://www.ticalc.org/


On Sat, 12 May 2001, Scott Noveck wrote:

>
> Think I'll start a new thread and see if I can get an answer to something
> that I've been wondering for a long time. . .
>
> I've noticed that if an 89/92+ is set with the complex mode to rectangular
> or polar, then it will return an imaginary answer when you try to take a log
> (any base?) of a negative number.
>
> For example, ln(-1) returns PI*i.
>
> I've been taking AP Calc AB this year, and I've yet to see anything about
> taking logs of negative numbers.  My precalc teacher last year insisted that
> it's absolutely impossible to do so -- she didn't say anything along the
> lines of it being impossible for us with our knowledge; she said it was
> outright impossible.  When I showed her the calc doing it, she had no clue
> what it was doing.
>
> I was surprised; I would expect that if someone is teaching math, they
> should at least understand concepts beyond their own knowledge.  Then again,
> since we Americans barely pay our teachers enough to make a living, they
> typically weren't the smartest ones back when they were in school.
>
> So I was wondering two things: at what level of math is this taught, and, if
> the concept is simple enough, could someone explain it?
>
> I think it's related to the formula e^(x*i) = cos(x)-i*sin(x), which I've
> seen online many times before but that I've yet to reach (does it come up
> with Taylor series?  I've seen it derived in context with them).  Anyways, a
> little enlightenment would be appreciated =)
>
>     -Scott
>
>
>
References: