Take
Here's the next one I could use some help on.
The principle argument of a complex number z, denoted by , is defined as the value of such that . Give an example of a complex sequence converging to a limit such that fails to converge.
I know of one such example:
This has a limit point of 0, but no well defined argument.
However the text says there is at least one other series with a limit point of -1. Not only have I been unable to construct an example (please help), but I can't see the logic behind this one at all. What is so special about -1 that, say, we can't construct one with a limit point of 1?
-Dan
I am having a slight problem with both TPH's and Plato's series. For both of them I am getting that and that's just not right, since the limit point is -1 giving an argument of . (Or is this the whole key to the problem?)
Additionally I can use both sequences to construct the same situation, but with any real number as the limit point. The text disagrees with this possibility:
Here's the whole problem statement:
My already disoriented mind is spinning...The principle argument of a complex number z, denoted by , is defined as the value of such that . Give an example of a complex sequence converging to a limit such that fails to converge. Show that this cannot happen unless or .
-Dan
The argument function defined by is continous on . This means if is a sequence of points not on and if they converge to then by the definition of continuity. This tells us any conterexamples to this problem has to be similar to how it was done above where the sequence of points jumps between and .
I'm trying to anyway. The text I am using is more of a review text, if you know what I mean, rather than a formal textbook. ("Complex Analysis With Applications" by Silverman.) I got it at Borders with 75% off. (It cost me a whole 3 bucks. ) Anyway, I figure I get understand most of it because I have taken the "Physics" version of it before. This will fill in many of the holes in my education, I hope. This particular chapter is a pain to me, though, since I have forgotten most of the series work I did in my Intro Calc classes.
Ah well. So I'm a sado-masochist.
-Dan