Re: strange 1
From: Hero (Hero.van.Jindelt_at_gmx.de)
Date: 09/12/04
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Date: 12 Sep 2004 08:19:59 -0700
Take another function, let's say cosinus:
It's defined for every real number r, for every r
You can compute or by geometrical means one can
find a f(r)=cos r.
Now, "Inverse function" - that's a holy wish, nearly
never comes true. Either it's not invers, or not a function
(for one s You can find several r's with f(r1)=s and f(r2)=s)
or there are numbers s with: there is no r with f(r)=s.
Now a mathematical plane, every point z has it's coordinates
z=(a,b) and the function e gives to every point z one f(z)=e^z
(e to the power of z)=e^((a,b))=e^(a+i*b)
e^((a,0))=e^(a) is a positive number,
e^((0,b))=e^(i*b)=(e^(b) )^(i), this is a direction,
to be more precise,it has unit 1 length and has the direction b
as it can also be expressed by
e^(i*b)= (cos b, sin b)
So, for every z one can find it's length and it's direction.
The "invers", the log, for a length and a direction find a
point z (with reference to the origin) with
e^z = length *(cos @, sin @)
There is the zero, which gives troubles, so exclude it in
Your definition.
And the @'s , the angels ?
You can look at angels as rotations, as movements - or as
difference of direction.
In the case of rotations, You can do several rounds, repeatedly
looking into the same direction, so one chooses only a
principal branch or one rotation (to keep it to a "function")
Or as difference of direction, one can have every angle from
zero to nearly 2*pi (but 2*pi itself is the same diretion as zero).
(A day makes 24 hours, but there's no 24 o'clock or ten minutes
past 24 o'clock - it' zero hours ten minutes)
Puh, finally You start with minus pi (but not the direction
itself), and crossing the zero coming to plus pi (but not
the point itself) then You exclude the negative numbers,
the left part of the x-axis.
Is there any reason, why one should do it ?
So, either include + pi or minus pi.
That is my opinion, and You have an opinion of one book
and an opinion of another book - which one is right,
You can find out Yourself.
Have fun
Hero
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