Author: vasco [ 15 Jun 2008, 07:29 ] Post subject: Exercise [05.10] b We know that generally (see my post on the general power )It’s important to make the distinction between 1) which is multi-valued and2) .... which is single valuedLooking at 1) above for z=1 we obtain Just as (in one of my earlier posts) I used Log w to represent the multi-valued logarithm of w, and log w to represent the principal value, such that , so I could use E(z) to represent the multi-valued 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 multi-valued and has the 2 possible distinct values -3 and +3.However, writing  and then sayingTherefore +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 multi-valued 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: RTRex5-10v2.pdf [35.03 KiB] Downloaded 646 times