# Complex Analysis homework questions (some are challenging)

1) Let U be a subset of C s.t U is open and connected and let f bea holomorphic function on U s.t. for every z in U, |f(z)| = 1, ie takes takes all points in U to the boundary of the unit circle. Prove that f is constant.

Pf.

Suppose f is not constant. Then we can find a w s.t. f'(w) is nonzero, else we know that f is holomorphic and hence has a power series representation, therefore since every w has zero derivative, f = a0, the first coefficient in the power series representation, ie f is constant. So let w be a point where f has nonzero derivative. We can therefore rewrite f(z) = f(w) + f'(w)(z-w) + h(z)(z-w) where h(z) --> 0 as z --> w, (WHY?, i dont recall why this is so, it's just in my notes for some reason). So f is nonconstant and analytic, therefore f is an open mapping. so consider some open neighborhood of w, call this neighborhood V, we know that f takes V to another open set f(V).

Here is where i am stuck, what gurantees that this point is not in the open neighborhood intersected with the boundary of the circle?????? i know it has something to do with us rewriting f(z) in terms of f(w) and h(z).

2) I guess this is a similar question, Suppose that f and g are holomorphic in a connected open set U and |f(z)|^2 + |g(z)|^2 = 1, show that f and g are constant.

Pf.
I have that u |--> (f(u), g(u)) which sits in C^2.
So converting to real coordinates, i get that: f(u) = x1 + iy1, g(u) = x2+iy2
so (x1)^2 + (y1)^2 + (x2)^2 + (y2)^2 = 1, giving me a 3 dimensional sphere sitting in R^4.

Now suppose that f and g are not constant, again, since they are holomorphic they are also analytic and also there exists a w in U with nonzero derivative, else f, g are constant functions.

So let's take a w with nonzero derivative and consider an open neighborhood around w, now both f and g are again open mappings, so it takes our open neighborhood around w, let it be denoted by V, to another open neighborhood around (f(w), g(w)). But again, how do i derive that the neighborhood of points contains a point that is not on the boundary of our sphere?????

3) Let f be an entire function, N a positive integer and C > 0
(a) Suppose that |f(z)| <= (less than or equal to) C|z|^N for every z in C. Show that f is a polynomial.

Pf.
I'm not very sure where to start on this one. I know that polynomials are entire functions. I'm guessing we want to show that if f has a power series representation, then after some positive integer G, for every n > G, an = 0?

(b) Suppose that |f(z)| => (greater than or equal to) C|z|^N for |z| large enough. Show that f is polynomial.

Pf.

eh not really sure again, sorry guys, im kinda stumped on this one.

(4)
Consider the function f(z) = 1 + z^2 + ...+ z^(2^(n-1)) + ..... and |z| < 1.

Show that for every a = e^(k*2pi*i/2^n), n greater than or equal to 0, k = 1, 2, ..., 2^n-1, that f(z) tends to infinity when z approaches a along the radius of the circle, lim (r>0, r--> 1 from the negative side) f(r*a) = infinity. Conclude that every point of the unit circle is a singularity for f.

Pf.
Hmm, well I first of all dont really even understand the question. We are saying, given an n and given an r that satisfies our limit condition then
f(z) --> infinity? and what exactly is a supposed to be???

(5) Suppose that f(z) is given by a power series and has radius of convergence equal to 1 and there are only poles of first order on the unit circle (no other singularities). Show that the sequence {an} is bounded.

Pf.

???? lol

(6) Suppose we wanted to represent the function

f(z) = 1/(1-z^2) + 1/(3-z) by a Laurent series. How many such representations are there? In which region is each of them valid? Find the coefficients aj explicitly for each of these representations.

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I know that there is more than 1 singularity, but they are isolated singularities. and there are 3 ways to represent a function by a Laurent series, one is by the power series if the function is continuous, however this is not a continous function, the 2nd is by by terms with negative indice coefficients, and the third is by an infinite series. Need some guidance on this one again.

Thank you!!!!

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4) There is a repeating pattern here after taking n many terms of the sum.

1) If you are allowed to use the open mapping property, what sort of sets are (not) subsets of image of f?

3) Using a famous integral formula, you can show that derivatives of sufficiently high order vanish identically. That is bounded in modulus by epsilon for any epsilon >0

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3) Using a famous integral formula, you can show that derivatives of sufficiently high order vanish identically. That is bounded in modulus by epsilon for any epsilon >0
im guessing you mean Cauchy's inequalities and use the fact that is an entire function?