3
votes
1answer
27 views

Relation between continuity of $f$ and analyticity of $f(z)^8$

If $f(z)$ is continuous on some domain $D$ and $f(z)^8$ (the function to the eighth power, not the eighth derivative) is analytic, then why does this imply that f is analytic on a neighborhood of each ...
0
votes
1answer
70 views

Is f(z)=1/z truly an analytic function

For an analytic function $f(z)$, we have $$\frac{\partial f}{\partial \bar{z}}=0.$$ Consider the function $f(z)=\frac{1}{z}$, which, at first sight, is a bona fide analytic function. However, we can ...
0
votes
1answer
26 views

Analytic Map from $B(0,1)$ to $B(0,1)$

Is the analytic map from $B(0,1)$ to $B(0,1)$ such that $f(0)=1/2$ and $f'(0)=3/4$ unique?
0
votes
0answers
22 views

Maximum modulus principle, is it true?

Suppose f is analytic in an open set containing the open disk D(2+3i, 7) and its boundary circle C(2+3i, 7) such that |f(z) + 7i + 24|<25 for all z in C(2+3i,7). Then f has no zeroes inside D(2+3i, ...
0
votes
0answers
20 views

Analytic Continuation of a Function Containing a Square Root to a Second Riemann Sheet

Consider the function $f(z) = g_1(z) + \sqrt{z} \, g_2(z)$, where $g_1(z)$ and $g_2(z)$ are entire functions, and we take the principal branch of the square root. $f$ is analytic on $\mathbb{C} / \{z ...
0
votes
0answers
29 views

Showing where complex function is analytic and differentiable.

I've been asked to show where the following function is analytic and differentiable; $$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$ First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$. Then, I ...
0
votes
0answers
34 views

Liouville's theorem for functions not bounded on an isolated set

On Liouville's theorem. Prove that if $f(z)$ is defined, analytic, bounded in the entire complex plane, except for an isolated set of points, then $f$ must be constant. What happens if said set has a ...
0
votes
1answer
108 views

Double exponential Taylor series $\exp(-\exp(k-ex))$

k is real constant $\gt = 1$. Is $a_n$ for $f(x)$ positive, increasing, and $\lt 1$, where $n\lt= e^{k-1}$? $$f(x) = \sum_{n=0}^{\infty} a_n x^n = \exp(-\exp(k-ex))$$ $f(x)$ is the double ...
1
vote
1answer
74 views

Proof of a result related to Liouville's Theorem

I was investigating the following corollary to Liouville's Theorem in Complex Analysis: if $f(z)$ is entire and $\lim_{z\rightarrow \infty}z^{-n}f(z)=0$, then $f(z)$ is a polynomial in $z$ of degree ...
0
votes
0answers
29 views

$f$ is analytic with range as a circle

I was given that range of $f$ lies on a cirlce, and $f$ is analytic on $D$. I want to show that $f$ is constant. This is my approach: I suppose that $f$ lies on a circle $|w-P|=R$, where $P,R$ are ...
1
vote
1answer
41 views

Complex Analysis analytic function

If $f$ is an analytic function on a domain $D$ and $\mathrm{Im} f$ takes on only the value $71$ then for some constant $C \in \Bbb{R}$, is it true that $f = C + 71 i$ on $D$?
0
votes
4answers
75 views

On the subject of holomorphic functions on an open disc, D.

The question I am pondering over is an interesting one: If $f(z) = u + iv$ is holomorphic on an open disc $D$, and the range of $f$ lies in either a straight line or a circle, prove that $f$ is ...
1
vote
2answers
65 views

Prove that the entire function $f$ is linear.

Suppose $f=u+iv$ be an entire function such that $u(x,y)=\phi(x)$ and $v(x,y)=\psi(y)$ for all $x,y\in\mathbb{R}$. Prove that $f(x)=az+b$ for some $a\in\mathbb{C},b\in\mathbb{C}$. My approach was: ...
1
vote
1answer
42 views

Complex Conjugation question

I had a complex analysis exam yesterday, and one of the questions is bothering me. Suppose $f(z)$ is an entire function. Show that $g(z) = (f(z^*))^*$ is also entire. Here $^*$ indicates complex ...
0
votes
1answer
42 views

Can we deduce that the zeros of $g$ are also isolated?

Let $f:Ω→ℂ$ be a non-zero holomorphic function and $g:Ω→ℂ$ be a non-zero non-holomorphic function. We know that all the zeros of $f$ are isolated. Assume that $$f(s)=0⇒g(s)=0$$ Can we deduce that the ...
0
votes
2answers
66 views

On Cauchy-Riemann equations

Given $f:\mathbb C\to \mathbb C$ is a non-constant entire function. Then which of the following is possible? Re $f(z)=$ Im $ f(z)$, Im$\,f(z)<0$, Re$\,f(z)$ is bounded, $f(z)\neq 0,$ for all ...
1
vote
0answers
22 views

Show that $f$ and $g$ are holomorphic in the set $D=({s=α+iβ∈ℂ: 0<α<1})$

Let us consider two complex functions $g,f$: $$g(α+iβ)=∑_{n=2}^{m}(-1)ⁿ⁻¹((n^{2α-1}-1)/n^{α})n^{iβ}$$ $$f(α+iβ)=(-1)^{m}(((m+1)^{2α-1}-1)/(m+1)^{α})(m+1)^{iβ}$$ My question is: Show that $f$ and ...
0
votes
1answer
42 views
0
votes
1answer
48 views

use example to prove the sum of two nonanalytic functions can be analytic [closed]

Find two functions, each of which is nowhere analytic, but whose sum is an entire function.
3
votes
4answers
260 views

If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic in a disk centered at the origin and $f$ satisfies the differential equation $$f'' = f.$$ Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for suitable constants ...
1
vote
1answer
24 views

Prove the following property of holomorphic functions.

Let $\rho(x)$ be a holomorphic function on a disk $D \subseteq \mathbb{C}$ with the property that $\rho(x) \notin \mathbb{N^*} = \{1,2,\dots\}$ on $D$. Prove the following: There exists an $R$ ...
3
votes
0answers
165 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
1
vote
0answers
17 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
1
vote
3answers
117 views

let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume $f(z) = f(2z)$, prove that f is constant

$f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume that $f(z) = f(2z)$ for all $z \in \mathbb{C}$. Prove that f is constant... Then we are supposed to use this result to ...
5
votes
4answers
223 views

Define a branch of $(z^2 − 1)^{1/2}$ which is analytic in the unit disk.

Hint: $z^2 − 1 = (z + 1)(z − 1)$. I'm really struggling with this question. I understand that for this function to be analytic it has to be differentiable in some neighbourhood, but I have no idea ...
1
vote
1answer
60 views

Why do u,v components in Cauchy-Riemann conditions are irrotational?

It's very strange to me! When we decompose a complex function to a real part and an immaginary part, we have $f(z) = u(x,y) + j v(x,y)$ following the conditions of analyticity we can derive the ...
2
votes
3answers
57 views

Corresponding analytic function?

I have found a general harmonic function of form $a x^3 - 3dx^2 y - 3axy^2 + dy^3$ and it's harmonic conjugate $v = 3ax^2y - 3dxy^2 + ay^3 + dx^3 + K$ where k is constant. I now am asked to find the ...
1
vote
1answer
81 views

Analytic continuation of complex square root

This example comes from the book by Elias Wegert: Visual Complex Functions. Consider the function $f(z):=z^{1/2}$ for $z \in \mathbb{C}$ with $|z - 1| < 1.$ For these $z$, the function can be ...
5
votes
1answer
65 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
1
vote
1answer
109 views

$f $ is analytic and maps the unit disk to itself. Prove that $|f'(0)|\leq1- |f(0)|^2 $

I am having difficulties with the following problem: $\bf Given$: $f $ is analytic and maps from unit disk to itself. $\bf Prove:$ $|f'(0)|\leq1- |f(0)|^2 $. For some reason (unclear to me) it ...
1
vote
1answer
32 views

When does analytic continuation respect functional equation

This is a subtle point about analytic continuation. Let $\Gamma(s)$ be the analytic continuation of $\gamma(s) := \int_0^\infty e^{-t}t^{s-1}dt$ to $\Bbb C \setminus \Bbb Z_{<0}$, the latter ...
2
votes
2answers
70 views

$f$ analytic and $|f|$ a function of $|z|$

Suppose $f$ is analytic inside the unit disc and that $|f(z)|$ depends only on $|z|$. Prove that we can write $f(z) = Cz^N$, for all $z$ in the disc. In the suggested proof, it is stated like it's ...
0
votes
2answers
49 views

Show that $\log|\sin(z)|$ is the real part of a holomorphic function

$D$ is a connected, simply connected domain with $\sin(z)$ never zero on D. Show that $\log|\sin(z)|$ is the real part of a holomorphic function. My question is: how to show $\sin(z)$ maps a simply ...
5
votes
1answer
87 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
0
votes
1answer
34 views

Analytic functions in two variables

Let $f$ be an analytic function in two complex variables. It is well known that we can expand $f$ in a convergent series of two variables. Can we separate the variables in such a manner that $f$ ...
2
votes
2answers
92 views

Behavior of holomorphic functions on the boundary of the unit disk

$\textbf{Problem.}$ Suppose $f$ is holomorphic on the unit disk $\mathbb{D}$. Show there are points $a_n\in \mathbb{D}$, $a\in \partial \mathbb{D}$, and $b\in \mathbb{C}$ such that $a_n\to a$ and ...
3
votes
1answer
54 views

Prove f is analytic and periodic

Suppose that there are entire functions $\{f_n\}$ so that for all complex numbers $x+iy$ $$\sum_{n=1}^{\infty} |f_n(x+iy)|^{\frac{1}{n}} \leq e^x$$ Show that $f(z)=\sum_{n=1}^{\infty} f_n(x+iy)$ is ...
1
vote
1answer
70 views

for two non zero complex polynomial $p(z),q(z)$ we have $p(z)\overline{q(z)}$ is analytic if and only if ?? CSIR - June $2013$

Question is : for two non zero complex polynomial $p(z),q(z)$ we have $p(z)\overline{q(z)}$ is analytic if and only if $p(z)$ is Constant $p(z)q(z)$ is Constant $q(z)$ is Constant ...
1
vote
1answer
60 views

The natural boundary of the modular $\lambda$ function is the real axis

I want to solve the following problem from Ahlfors' text: Show that the function $\lambda(\tau)$ introduced in Chap. 7, Sec. 3.4, has the real axis as a natural boundary. Here $\lambda(\tau)$ is ...
4
votes
1answer
87 views

(solution verification) the series $\sum z^{n!}$ has the unit circle as a natural boundary

I've tried to solve the following problem from Ahlfors' complex analysis text: If a function element $(f,\Omega)$ has no direct analytic continuations other than the ones obtained by restricting ...
0
votes
1answer
36 views

Specify analytic function

How can I check that function $f(z)=z^3+z-1$ is analytic or not without Cauchy-Riemann equations? $(z\in\Bbb C)$
0
votes
0answers
20 views

Is $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ a general analytic function?

I have an expression $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ where $c(k)\in\boldsymbol{C}$ and $k\in\boldsymbol{R}$. $f(z)$ is an analytic function, since it contains only non-negative ...
3
votes
1answer
77 views

Complex Analysis: Log Function

I want to approach this problem with maximum understanding of everything that is going on. I have the function $F(z)=\log(z^2+4)$, and I want to give a region in which it is analytic. I guess I ...
0
votes
1answer
36 views

Proving that a metric on space of analytic functions is equivalent to compact convergence

Let $U\subseteq \mathbb C$ be open and $\mathscr A(U)$ consist of all analytic functions on $U$. I can easily prove that there exists a sequence $K_n$ of compact sets in $U$ so that ...
5
votes
1answer
92 views

Proof of the three-point characterization of holomorphy

This post on Math Overflow is looking for the source of the following theorem: Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ denote the open unit disk. A function $f : D \to D$ is holomorphic iff ...
1
vote
1answer
51 views

Specifying a holomorphic function by a sequence of values

Given a sequence $(z_n, w_n)$ of pairs of complex numbers such that $|z_n| \to \infty$ as $n \to \infty$, there exists a holomorphic function $f$ such that $f(z_n) = w_n$ for all $n$. Proof: By the ...
1
vote
0answers
26 views

When is meromorphic continuation possible?

Suppose I have an expression of the form $$f(z) := f_1(z)+f_2(z)$$ ($f,f_1,f_2$ can e.g. be integrals) with $f_1$ convergent in the region $R_1=\{\Re(z)>-1\}$ and $f_2$ convergent in the region ...
0
votes
1answer
41 views

Principal Logarithmic Question

Here is a question that is driving me insane: Show that $p.v \sqrt{z-1}\times p.v\sqrt{z+1}=-p.v.\sqrt{z^2 -1}$ for $Re(z)<-1.$(p.v. stands for the principal singular valued logarithmic ...
2
votes
1answer
88 views

To what extent is a function that is analytic on the unit disk determined by its boundary values?

Suppose we have a function that is analytic on the open unit disk. Suppose we have a continuous function on the boundary of the disk that maps each point on the boundary of the disk to its conjugate. ...
1
vote
1answer
122 views

Composition of continious and analytic map

Let $U,V,W\subset\mathbb{C}$ open and connected, $f:U\to V$ continous and $g:V\to W$ analytic and non-constant. If $g\circ f$ is analytic, does then $f$ have to be analytic as well? I guess the ...