Help with complex analysis problem.

[Meital]

Can someone help me with this problem. I need any hints to help me get started. I am assuming limit points of cisx are -1- i and 1+i I am not sure if this is correct. And if it, I don't know how to show that it is dense, I know the def of a dense subset, but I don't know how to apply it here. Please help me.

 

Show that {cis K : K is a non-negative integer) is dense in T = { z in C ( Complex space) : |z|= 1}. For which values of theta is {cis ( k*theta) : k is non-negative integer} dense in T?

[ElijahJones]

Well T here is just the unit circle (right its all points one unit from the origin). Lets cal lyou set M. M is dense in T means that the closure of M is T. So you want to show that among points one unit from the origin, the closure of M is T. What is the closure? Its the intersection of all closed sets containing M in T. It is the union of M and its limit points in T.

 

I cannot find your function cis(x) even in Knopp. But suppose the limit points are in fact {-1-i,1+i}. Well look these cannot be the limit points because [MATH]||-1-i||^2=2[/MATH] so your limit points are wrong. Anyways I would be happy to help further. The outline of the prooof is not difficult. What exactly is your function cis(x) and how is it constrained to T?

[Meital]

cis k = cos k + i sin k

[matt grime]

it only mkes snese to speak of a limit point of a fucntion of x if you are saying where x tends to. in this case cos(x)+isin(x) is our fucntion. as x tends to what? as it

 

in this case it is better not to 'invent' new notation (i have never in many years of doign maths at universities in two countries met the term cisx for what yoy describe especially as it is just exp(ix)

 

so, you are asking for a proof that exp(ik), k in N is dense in the unit circle of C. well, that is becuase it is an irrational rotation of the circle and all orbits are dense. look it up in any basic introdcution to dynamical systems.

[Meital]

Actuallty this Q is from a book called " functions of one complex variable I" 2nd edition.

[matt grime]

i did put 'invent' in inverted commas as i knew you were using something you'd seen in a textbook once, a textbook that no one else here may have read.