
Complex Analysis Complex Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
December 5th, 2017, 04:24 AM  #1  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,396 Thanks: 100  Fundamental Theorem of Algebra Proof 02 Quote:
w=e$\displaystyle ^{z}$ was offered as contradiction because assertedly w=0 doesn't exist. w=0 is perfect example of isolated point of the Theorem above. Consider the rays in w heading toward the origin: $\displaystyle w=Re^{i\Theta}, \Theta = constant, R \rightarrow 0$. The corresponding "curve" in z is $\displaystyle e^{z}= Re^{i\Theta}$, or $\displaystyle e^{x}=R$ and $\displaystyle y=\Theta$ =constant. $\displaystyle R \rightarrow 0$, if $\displaystyle e^{x} \rightarrow 0$, which it does if $\displaystyle x \rightarrow \infty$, and the limit exists, ie, limit point w=0 of the line approaching origin exists, conditions of the theorem above are satisfied, and e$\displaystyle ^{z}$ maps to all of w. If you are talking about maps from all of z to all of w, you have to deal with infinity, ie, possibility of existence of limits as something gets unboundedly large.  
December 5th, 2017, 05:38 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,341 Thanks: 2463 Math Focus: Mainly analysis and algebra  No, because $\infty \not \in \mathbb C$. If something gets unboundedly large, it has no limit by definition. Moreover, the limit is not part of the range of a function. The real function $f(x)=\frac1x$ with domain $D=\{x \in \mathbb R: x \gt 1\}$ has, as it's range, the open interval $(0,1)$. It does not include either $0$ or $1$ even though they are limits because the function never attains those values.


Tags 
algebra, fundamental, proof, theorem 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Fundamental Theorem of Algebra Proof  zylo  Complex Analysis  50  December 5th, 2017 08:10 AM 
Fundamental Theorem of Algebra Proof 01  zylo  Complex Analysis  15  November 30th, 2017 10:39 AM 
proof of fundamental theorem of arithmetic  AspiringPhysicist  Abstract Algebra  3  October 2nd, 2014 12:19 AM 
Is this proof of the fundamental theorem of arithmetic ok?  king.oslo  Algebra  2  September 19th, 2013 10:48 PM 
Fundamental Theorem of Algebra  johnny  Algebra  11  December 21st, 2007 09:51 PM 