A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers is quite different from the properties ...
2
votes
1answer
150 views
Evaluating $f(z)=\sqrt{z^2-1}$, given the branch I am on.
I'm working on a problem in Gamelin's Complex Analysis (Chapter IV, Section 2, page 109, exercise #4). I'm asked to consider the branch of $f(z)=\sqrt{z^2-1}$ on $D=C\setminus (-\infty,1]$ that is ...
3
votes
1answer
79 views
Basic question about analyticity vs. differentiability in complex analysis.
In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula,"
3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
0
votes
3answers
43 views
some proofs in complex related to F.T of algebra and cauchy's inequality maybe
these two questions i didn't even find the way to solve
so please if you can help
1)suppose f(z) is entire with |f(z)| <= |exp(z)| for every z
i want to prove that f(z) = k exp(z) for some |k| ...
2
votes
1answer
61 views
Analyticity of $\frac{Log(z+4)}{z^2+i}$
This problem is from Churchill and Brown. How do I prove that
$f(z)=\frac{Log(z+4)}{z^2+i}$ is analytics everywhere except $\pm\frac{(1-i)}{\sqrt{2}}$ and on the portion $x \le -4$ of the real axis.
...
3
votes
2answers
41 views
Determining whether a family of power series is normal
How should I check whether a given family of power series forms a normal family? I am trying to apply Montel's theorem that says that a family of holomorphic functions is normal iff it is uniformly ...
0
votes
1answer
71 views
in complex analysis need some examples about uniform convergence and analyticity
I am new in solving anything in complex
and I am stuck on two examples :
1) I read that : The limit of the sequence of non-analytic functions converging uniformly inside a simple closed curve can be ...
3
votes
1answer
57 views
Notation in Old Paper
I am reading a paper
http://www.ams.org/journals/bull/1935-41-04/S0002-9904-1935-06049-5/S0002-9904-1935-06049-5.pdf
and I am wondering about the notation in the lemma page which states that
...
0
votes
0answers
50 views
Internal point transformed in an external one?
Let $f \colon \Omega \to \mathbb{C} $ be an analytic function over a connected open subset $\Omega$ of $\mathbb{C}$ and let $\gamma$ a rectifiable closed curve in $\Omega$. If $a$ is a point which is ...
1
vote
2answers
69 views
How to solve this problem in detailed steps?
If $f(z)$ is analytic, prove that $$\left(\frac{d^{2}}{dx^{2}}+\frac{d^{2}}{dy^{2}}\right)\left|f\left(z\right)\right|^{2}=4\left|f^{\prime}\left(z\right)\right|^{2}$$
0
votes
1answer
35 views
Show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$
This is a homework problem.
For $|z| \le R$ and $|a_j| < R$ for $j=1,\ldots, k$, not all zero, show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$.
...
2
votes
0answers
88 views
Zeros of analytic function and limit points at boundary
Let $S$ be the open ball of center $0$ and radius $1$ with $0$ removed in the complex plane. Is the function $f(z)=\sin(1/z)$ a valid example of analytic function defined in an open subspace whose ...
5
votes
2answers
91 views
$\lim_{x\rightarrow\infty}\frac{f(x)}{e^x}$ for analytic functions
For some analytic function $f(x)=\lim_{n\rightarrow\infty}\sum^n_{r=0}c_rx^r$,
...
2
votes
0answers
65 views
Prescribing zeroes, poles, principal parts and finitely many terms with positive exponents in Laurent series
I was given a problem to prove a theorem by Mittag-Leffler about prescribing the items in the title, using Weierstrass's theorem about prescribed zeroes and Mittag-Leffler's theorem about prescribed ...
0
votes
2answers
51 views
Finding analatical function
As far as i know that in differential equations an analytical function can be represented in terms of the power series and also using power series we can always determine such an analytical function. ...
11
votes
1answer
287 views
Images of compact subsets in the plane
Let $K$ be an infinite compact subset of $\mathbb{C}$. Is it true that there exists a sequence $(f_n)_{n>0}$ of functions holomorphic in some neighborhood of $K$, such that the images $f_n(K)$ are ...
4
votes
2answers
76 views
Analytic off the real axis
If $f:\mathbb C \longrightarrow \mathbb C$ is continuous and $f$ is analytic off the real axis, then show that $f$ is entire.
0
votes
0answers
24 views
Linearly dependent analytic functions [duplicate]
Possible Duplicate:
Problem on exponential of entire function
If f(z) and g(z) are entire functions such that $$
e^{f(z)},e^{g(z)}, {1} $$ are linearly dependent.
That is there is ...
4
votes
1answer
102 views
Mean values theorem and countable sets
The mean values theorem says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u)$$ My question is: Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u)$$ Assume that $f$ is a ...
1
vote
0answers
112 views
Schwarz reflection principle and bounded derivatives
Suppose $f$ is a holomorphic function on $\Omega^+$ (an open subset of the upper complex plane) that extends continuously to $I$ (a subset of $\mathbb{R}$). Let $\Omega^-$ be the reflection of ...
3
votes
1answer
85 views
Domain of bijectivity of function $f:\mathbb{C}\rightarrow\mathbb{C}$
There is a type of problems in my course in Complex analysis that I don't fully understand them.
Given function $f:\mathbb{C}\rightarrow\mathbb{C}$, $f(z)=z^2$. You must specify the analytic and ...
7
votes
2answers
252 views
True/False Questions for Complex Analysis
I am studying for my (introductory) complex analysis final exam tomorrow. I am practicing an old final exam, which unfortunately has no answer key. Here is a link:
...
11
votes
2answers
242 views
Holomorphic function on the unit disk $f$, show the set $z,w\in \mathbb{C}$ such that $f(z)=f(w)$ is not countable
Here is the problem statement:
Suppose $f$ is a holomorphic function on the unit disk. Show that the set $A=\lbrace (z,w) \in \mathbb{C}^2\;|\; |z|,|w| \leq \frac{1}{2}, z\neq w, f(z)=f(w)\rbrace$ is ...
3
votes
1answer
134 views
Representation of holomorphic function on punctured unit disk [duplicate]
Possible Duplicate:
Non zero analytic functions on annulus
Let $f$ be holomorphic on the punctured unit disk $\{z\in\mathbb{C}:0<|z|<1\}$ and suppose $f$ has no zeros. I want to show ...
1
vote
2answers
95 views
Convolution and analyticity
Assume $f$ and $g$ are continuous and related to each other as
$$
f(x) = \int _{0}^{x-1} \Big ( (x- y)^2 - 1\Big )^{3/2}g(y) \, dy, \qquad x>1.
$$
If we happen to know that $f$ is real analytic ...
0
votes
1answer
135 views
Complex function defined as an integral: is it analytic?
I'm trying to study for my final this Thursday and for some reason the problem that is giving me the most trouble is, according to the professor, the "easiest one". I'm just not seeing something here, ...
4
votes
1answer
124 views
composition of non analytic functions can be analytic?
I was wondering if the following scenarios are possible:
1) composition of an analytic function with a non-analytic continuous function being analytic (except for trivial cases)
2) composition of two ...
0
votes
1answer
108 views
Extension of Liouville's Theorem?
Liouville's Theorem states that if a function is bounded and holomorphic on the complex plane (i.e. bounded and entire), then it is a constant function.
What if we consider the following, slightly ...
1
vote
4answers
225 views
Proving $\sqrt{2z-2\log(z)-2}$ is analytic near $z=1$.
I am trying to prove $f(z)=\sqrt{2z-2\log(z)-2}$ is analytic near $z=1$. The issue is proving there is no branch point.
If I try the approach of taking the path $z=1+r\exp(i\theta)$ with $r=\epsilon$ ...
3
votes
0answers
176 views
Question Relating with Open Mapping Theorem for Analytic Functions
This problem is taken from Section VIII.4 of Theodore Gamelin's Complex Analysis:
Let $f(z)$ be an analytic function on the open unit disk $\mathbb{D}=\{|z|<1\}$. Suppose there is an annulus ...
1
vote
1answer
119 views
Defining branch of $\log z$ in annulus such that $|\log z|$ is unbounded
I am trying to construct a nonempty open subset $D$ of the annulus $\{z:1<|z|<2\}$ such that (i) $D$ is connected and so is its boundary, (ii) a holomorphic branch of $\log z$ can be defined on ...
1
vote
0answers
34 views
How to show that $f(x,y)$ are real analytical
Given a real function $f(x,y)$ on $R^2$, we know that (from wiki) it is called real analytic if it is locally given by a convergent power series.
My question is that whether there has some principle ...
2
votes
1answer
239 views
Problem based on Schwarz Lemma
Let $D={z\in\mathbb{C}:|z|<1}$ and $f:D\to D$ be analytic with f(0)=0
(i) Show that $|f(z)+f(-z)|\leq2|z|^2$
(ii) Suppose that $|f(z_0)+f(-z_0)=2|z_0|^2$ for some $z_0\in\mathbb{C}\setminus{0}$. ...
2
votes
1answer
159 views
Riemann Zeta Function Manipulation
The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$
(i) show that for $Re z> 1$, we have $$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty ...
3
votes
1answer
140 views
What are the analytic isomorphisms of $\Omega = \mathbb{C} \setminus \{p_1,\ldots,p_n\}$?
By an analytic isomorphism of $\Omega$, I mean an analytic function $\Omega \to \Omega$, with an analytic inverse.
2
votes
1answer
162 views
Show that an analytic function in the unit disc satisfies a inequality
Question: Let $f$ be an analytic function in the unit disc $D={z\in C: |z|<1}$. Consider a point $z_0\in D$. Show that there must be a positive integer n such that the n-th derivative of $f$ at ...
0
votes
1answer
214 views
Show that if $f$ is analytic in the unit disc then an integer $n$ such that $f(1/n)$ does not equal $1/(n+1)$
This is a variant of question Show that if f is analytic in $|z|\leq 1$, there must be some positive integer n such that $f(\frac{1}{n})\neq \frac{1}{n+1}$.
(i). Show that if $f$ is analytic in the ...
0
votes
1answer
159 views
A consequence of Runge's theorem
I'd like to have a reference for the proof of the following fact of complex analysis. I think it follows from Runge's theorem, but I don't know how to prove it.
Fact. Let $U \subseteq V \subseteq ...
4
votes
2answers
60 views
Number of times two rescaled, 'fully' monotonic functions can cross
Consider two functions $f: [0,1) \rightarrow \mathbb{R}$ and $g: [0,1) \rightarrow \mathbb{R}$. Suppose $f(x) > g(x)$ for all $x \in [0,1)$. Suppose further that $f$ and $g$ are infinitely ...
3
votes
3answers
211 views
The distinction between infinitely differentiable function and real analytic function
I have known that all the real analytic functions are infinitely differentiable.
On the other hand, I know that there exists a function that is infinitely differentiable but not real analytic. For ...
1
vote
1answer
93 views
Principal ideal ring analytic functions
Could someone sketch a proof and explain me in words, why the set of analytic functions on $\mathbb{C}$ does not form form a principal ideal ring?
Thank you!
2
votes
1answer
127 views
Ring of analytic functions on the circle
Let $A = C^\omega(S^1)$ (resp. $C^\omega_{\mathbb C}(S^1)$) the ring of real-analytic real-valued (resp. complex valued) functions on the circle.
These rings have maximal ideals $\mathfrak m_p = ...
13
votes
2answers
305 views
Can a function “grow too fast” to be real analytic?
Does there exist a continuous function $\: f : \mathbf{R} \to \mathbf{R} \:$ such that for
all real analytic functions $\: g : \mathbf{R} \to \mathbf{R} \:$, for all real numbers $x$,
there exists ...
5
votes
2answers
226 views
Do all analytic and $2\pi$ periodic functions have a finite Fourier series?
Consider a function $f:\mathbb{R}\to\mathbb{R}$ which is periodic with period $2\pi$. Let us impose the condition that $f$ is analytic. Now does that imply that $f$ has a finite Fourier series?
PS : ...
1
vote
0answers
58 views
Calculating germs for complete holomorphic function
I'm trying to find the germs $[z,f]$ for the complete holomorphic function $\sqrt{1+\sqrt{z}}$ . The question indicates that I should find 2 germs for z = 1, but I seem to be able to find 3! Where ...
2
votes
1answer
89 views
A Paley-Wiener like theorem in real-analysis
I try to identify conditions for the Fourier-transformation $\mathcal{F}(f)$ of some function $f \in L^1(\mathbb{R}^n)$ to be real-analytic. Namely I want to show that one of the following two ...
0
votes
1answer
173 views
Analytic non constant function
Stuck up on something in complex analysis.
Let $f$ analytic function and open $\Omega \subset \mathbb{C}$. Show that if $f$ is not a constant on a neighbourhood of $z_0$, then exist a neighbourhood ...
1
vote
1answer
440 views
If $f$ is a non-constant analytic function on a compact domain $D$, then $Re(f)$ and $Im(f)$ assume their max and min on the boundary of $D$.
This is a homework problem I got, my attempted proof is:
Since $f$ is non constant and analytic, $f=u(x)+iv(y)$ where neither $u$ nor $v$ is constant(by Cauchy Riemann equations) and $u v$ are both ...
2
votes
0answers
158 views
Area and locally one-to-one analytic mappings of the unit disk.
We learned about conformal mappings and various properties of locally one-to-one, analytic mappings of the unit disk. I am having trouble with the following problem, can anyone help?
Let $f(z) ...
6
votes
3answers
218 views
Expressing the area of the image of a holomorphic function by the coefficients of its expansion
I have the following problem.
Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...
4
votes
1answer
111 views
Entire function invariant on the coordinate axes (as sets).
From old qualifying exam: Let $E$ be the union of the two coordinate axes, i.e. $E = \{z=x+iy : xy=0\}$. Describe all entire functions satisfying $f(E) \subset E$.
I feel like the best approach is to ...
