# holomorphic

1. ### Complex Analysis - f is holomorphic...

Hey guys. Heres my problem: f is holomorphic on C and of the form f(x + iy) = u(x) + i v(y) where u and v are real functions, then f(x) = \lambda z + c with \lambda \in R and c \in C. Where C is the coplex numbers and R is the real numbers. Earlier I've made an assignment where I show that if...
2. ### Holomorphic functon

The following is a multiple choice question in which more than one option may be right. Let D={z belongs to C: |z| <1} which of the following is true? a) There exists a holomorphic function f from D->D with f(0)= 0, f'(0)=2 b) There exists a holomorphic function f from D->D with f(3/4)= 3/4...
3. ### Cauchy-Riemann equations and holomorphic functions

Consider the function defined by f(x+iy) = \sqrt(\left |x \right |\left |y \right |). Show that f satisfies the Cauchy-Riemann equations at the origin yet it is not holomorphic at zero. I can show that it satisfies the C-R equations but unsure about showing it is not holomorphic at zero...
4. ### Holomorphic function theorem

My complex analysis book states this as a theorem without proof: "A holomorphic function f(z) cannot map a ball D (subset of) C into a unit circle S1 (subset of) C unless f is constant." It's in the chapter on the Cauchy Riemann Equations. Maybe it's supposed to be obvious, but I don't get it...
5. ### Simple question about a holomorphic function

Proof: if f holomorphic then f(z)=λz+c Hello. I need a bit of help or a tip maybe.. I am to show that if f is a holomorphic function that is of the form f(x+iy) = u(x) + i*v(y) where u and v are real functions, then f(z) = λz+c where λ is a real number and c is a complex one. How would I...
6. ### example holomorphic function series

hello, firstly i'm sorry because my english isn't well , i hope i can describe the task correctly. I have to find a holomorphic function series f_n on E:={z \in IC; |z| <1 } such that f_n convergences compact to a not-constant function f, at least every function of f_n has one root on E, but f...
7. ### Why holomorphic ?

I cannot understand the following statement in the proof of Lemma 8.7 in "Lectures on Riemann Surfaces" by Otto Forster. where D(R)=\{z\in C : |z|<R \}, \;\;\; R>0, F(z,w)=w^n+c_1(z)w^{n-1}+\cdots+c_n(z), c_1(z)\cdots c_n(z)are holomorphic functions on D(r). In the equation...
8. ### Holomorphic Extension

Hello. I've been trying to show that the function \displaystyle f(z)=\sum_{n=1}^\infty \frac{z^{n!}}{n} cannot be holomorphically extended outside the unit disk D(0,1). I'd like to show that the function "blows up" at the boundary, but it isn't clear to me how to do that, except for simple...
9. ### Holomorphic function on a unbounded domain

I have problem in understanging the following question Q. Show by means of an example that a non-constant holomorphic function on a n unbounded domain need not achieve it's maximum modulus on the boundary of that domain Answer: here we assume that tha boundary is not empty. Maximum Modulus...
10. ### Injectivity of Holomorphic function

Let U be an open disk around the origin in \mathbb{C}. Suppose f:U \rightarrow \mathbb{C} is holomorphic on U ,f(0) = 0 and f'(0) = 1. I want to show that there exists a neighborhood V of 0, V \subset U, so that f is injective on V. Anybody can help?
11. ### Holomorphic function

Hey who can help me with this: Let f be holomorphic in a neighborhood of a closed disk \bar{D}. Then g(z) = \int_{\delta D}\frac{f(\zeta)d\zeta}{\zeta -z} is a holomorphic function on \mathbb{C}-\bar{D}. Why, and which one?
12. ### SOLVED Area of a holomorphic function's range on the unit disk

Hello all. Suppose f\in H(D(0,1)), where D(0,1) is the open unit disc, f is one-to-one in D(0,1), \Omega=f(D(0,1)), and f(z)=\sum c_{k}z^{k}. Prove that the area of \Omega is \pi\sum_{n=1}^{\infty}n|c_{n}|^{2}. ........................................... Here are my partial solutions...
13. ### Schwarz lemma and holomorphic function

Hi, I've been trying to figure out this problem for ages and am stuck. The question is: if f is holomorphic on the open unit disk and continuous on the closed unit disk, fixes the origin (f(0)=0) and |f(z)|=<|e^z| for |z|=1, then what is the max value of some z_0. My answer so far: The...
14. ### Bergman space of holomorphic functions on unit disc is closed in L^2 (unit disc)

Hello! Could you please help with the following question: Let A^2 (\mathbb{D} ) be the Bergman space of all holomorphic functions on the unit disc \mathbb{D} which also belong to L^2 ( \mathbb{D}) . Let f \in A^2(\mathbb{D}), \ 0<s<1 , and |z| < s. Cauchy's Integral Formula gives...
15. ### Is the conjugate of a complex function holomorphic?

Hello Suppose f(z) is holomorphic on the open disc D(0,1) of the complex plane. How would one go about proving g(z) = \overline{ f( \overline{z} ) } is holomorphic in D(0,1)? I attepted this by writing out the definition of the derivative, substituting \overline{ f( \overline{z} ) } =...
16. ### About the Collatz Conjecture, Complex Maps and Holomorphic Functions

First of all, I wasn't quite sure of where to post this thread, if it is wrongly located please forgive me. Just out of curiosity I've been pondering over some holomorphic functions, for generating fractals, associated with the Collatz Conjecture. Now I've come up with these complex maps which...
17. ### Holomorphic functions with constant modulus

Problem Statement: Suppose that f is holomorphic on a domain \Omega \subseteq \mathbb{C} with |f| constant. Prove that f is a constant map. Ideas: If |f| = 0 then f = 0. Suppose |f| = b \in \mathbb{C}, b \ne 0. So f(z) = b e^{i \theta(z)} where \theta(z) = arg_0(f(z)) with arg_0(f(z)) is a...
18. ### holomorphic function

A proof (of Morera's Theorem) ends with the following paraphrased line: **nevermind** proof continued on following page, i misread... by the way, anyone know how to delete a post you yourself posted that has become irrelevant?
19. ### Holomorphic extension of Dirichlet Series

Define L : \{s \in \mathbb{C}| Re(s) > 0\} \rightarrow \mathbb{C} by L(s) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^s}. Note that L(s) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}}{e^t+e^{-t}}dt Show that L can be extended to an entire function L : \mathbb{C} \rightarrow \mathbb{C}
20. ### Holomorphic Function

Let f : open unit ball -> C be continuous. Suppose that f is (complex)differentiable for all z in the unit ball with f(z) not equal to 0. Prove that f is holomorphic. The question reduces to simply the case if for all balls around z0 with f(z0)=0, we have some zero and non-zero f(z). The other...