The Road to Reality http://www.roadtoreality.info/ 

Exercise [05.10] b http://www.roadtoreality.info/viewtopic.php?f=19&t=167 
Page 1 of 1 
Author:  vasco [ 15 Jun 2008, 07:29 ] 
Post subject:  Exercise [05.10] b 
We know that generally (see my post on the general power ) Itâ€™s important to make the distinction between 1) which is multivalued and 2) .... which is single valued Looking at 1) above for z=1 we obtain Just as (in one of my earlier posts) I used Log w to represent the multivalued logarithm of w, and log w to represent the principal value, such that , so I could use E(z) to represent the multivalued exp (z) and e the principal value: , where k is any integer value, positive or negative. if we take k in the range 2 to +2 then we get: (the e we all know and love!) These are all separate valid values for Writing as is done in Ex 05.10, is not a valid thing to do. It is the same as saying: And therefore . A simpler, more obvious example of a similar fallacy would be for example sqrt(9). This is also multivalued and has the 2 possible distinct values 3 and +3. However, writing and then saying Therefore +3=3 is clearly fallacious. Or, another example but we know therefore fallacious again. When we say, for example above, that +3=3 because they are both equal to sqrt(9), we are forgetting that sqrt(9) is a multivalued function and therefore we are not allowed to equate the two values. This is exactly the same as saying is equal to . If we take logarithms then we can see the fallacy immediately: and it is clear that 
Author:  vasco [ 09 Aug 2008, 15:40 ] 
Post subject:  Re: Exercise [05.10] b 
Rather than edit my previous post on this exercise, which I believe to be essentially correct, I would like to submit the following, which I think gives a better explanation. Attachment: 
Author:  Azrael84 [ 22 Sep 2011, 13:07 ] 
Post subject:  Re: Exercise [05.10] b 
One thing I don't understand here is: yes e and e^{1+2iPi} are just two values of the multifunction E(1) so you can't just go e=E(1)=e^{1+2iPi} so therefore e=e^{1+2ipi} isnt true necessarily anymore so than you can go 3=sqrt(9)=+3 so 3=+3. But in the case of E(1) the two apparently distinct values are actually equal aren't they because of Euler ident? It's like having a mulifunction f(x) which has "distinct" values +x and (x) or something because e^{1+2iPi}=e^1e^{2iPi}=e^1 (cos(2Pi)+isin(2Pi)=e^1? I'm obviously confused about something here.. I think the key is probably you mean e in the series sense at the end? i.e e^z=1+z+z^2/2+..... then e^1=1+1+1/2+..... and e^{1+i2Pi}=1+1+i2Pi+1/2(1+i4Pi4Pi^2}+..... so in that sense the multivalues are not equatable? 
Page 1 of 1  Archived: 07 Aug 2014 
Powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group http://www.phpbb.com/ 