Complex Analysis homework questions (some are challenging)

In summary, the conversation discusses various proofs and explanations for different problems involving complex analysis. The first problem proves that a holomorphic function that maps all points in a connected open set to the boundary of the unit circle is constant. The second problem is similar, showing that two holomorphic functions with a certain property are constant. The third and fourth problems involve entire functions and their properties, while the fifth problem involves a function with isolated singularities. The sixth problem deals with representing a function as a Laurent series.
  • #1
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 don't 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, I am 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 don't 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.

-
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 continuous 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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  • #2
4) There is a repeating pattern here after taking n many terms of the sum.
 
  • #3
1) If you are allowed to use the open mapping property, what sort of sets are (not) subsets of image of f?
 
  • #4
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
 
Last edited:
  • #5
gammamcc said:
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?
 

1. What is the purpose of complex analysis in mathematics?

Complex analysis is a branch of mathematics that studies functions of complex numbers. It provides tools and techniques for understanding the behavior of these functions, which have applications in many areas of mathematics, physics, and engineering.

2. How do I approach solving a challenging complex analysis homework problem?

Solving a challenging complex analysis problem requires a systematic approach. Start by understanding the problem and identifying the key concepts and techniques that may be relevant. Then, break the problem into smaller, more manageable steps and use your knowledge of complex analysis to solve each step. Don't be afraid to ask for help or consult additional resources if needed.

3. What are some common techniques used in complex analysis?

Some common techniques used in complex analysis include power series, Laurent series, Cauchy's integral theorem, and the residue theorem. These techniques can be applied to evaluate integrals, find solutions to differential equations, and study the behavior of complex functions.

4. Can complex analysis be used to solve real-world problems?

Yes, complex analysis has many applications in real-world problems. For example, it is used in fluid dynamics to model the flow of fluids, in electrical engineering to analyze the behavior of electric circuits, and in finance to study the behavior of stock prices.

5. Are there any tips for improving my understanding of complex analysis?

Practice, practice, practice! Complex analysis can be a challenging subject, so it's important to work through lots of problems to improve your understanding. It can also be helpful to visualize complex functions and their behavior using tools such as complex graphs or contour plots. Finally, don't hesitate to seek help from your professor or classmates if you are struggling with a particular concept or problem.

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