Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [holomorphic-functions]

For questions on holomorphic functions, complex-valued functions of one or more complex variables that are complex differentiable in a neighborhood of every point in its domain.

2
votes
1answer
35 views

Why needed Open disk in domain?

Why is in above theorem, it is assumed that $D$ an open disk? Is it to make sure that we can surely apply mean value Theorem, ie, walking parallel to x or y axis, we are not moving out of domain? Can ...
0
votes
0answers
23 views

CR holomorphic functions

Let $\Omega \subset \mathbb{C}$ be a domain, $\mathcal{O}(\Omega)$ denote holomorphic functions on $\Omega$ and $\mathcal{C}^{\infty}(\overline{\Omega})$ functions smooth up to the boundary. I'm ...
1
vote
2answers
21 views

Existence of a bounded function on the disk with specified values

I am trying to answer the question: “Does there exist a bounded holomorphic function on the unit disk such that for all $n\in \mathbb{N}$, $f(1-\frac{1}{n}) = \frac{(-1)^n}{n}$?” My solution goes ...
3
votes
2answers
37 views

A function holomorphic on $\mathbb{D}(0,2)$ and bounded on a unit circle is a polynomial

Suppose function $f(z)$ is holomorphic on $\mathbb{D}(0,2)$ and $N>0$ is an integer such that: $$ |f^{(N)}(0)| = N! \sup\{|f(z)|: |z|=1\} $$ show that $f(z) = cz^N$, $c \in \mathbb{C}$. I have ...
0
votes
1answer
34 views

Questions About the Proof of Cauchy–Pompeiu Integral Formula.

I am studying function theory in several complex variables and the book I am using is "Tasty Bits of Several Complex Variables" by Jiří Lebl: https://www.jirka.org/scv/scv.pdf. At the moment I am ...
1
vote
1answer
123 views

Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.

Verify that $$\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$ and $r > 0$. I'm stuck. here is my attempt: $|...
-1
votes
2answers
50 views

Möbius transformation/biholomophic funtion

I have to show, that the Möbius transformation $$ T(z) = \frac{z-z_0}{1-\bar{z_0}z}$$ is an biholomorphic function on $ \mathbb{D}$. $ \mathbb{D}:=\{ z \in \mathbb{C}: |z|<1 \}$ and $z_0 \in \...
1
vote
3answers
70 views

When is a real-valued continuous function on the unit circle the real part of a holomorphic function on the unit disk?

This is known, but it is taking me some time to find the relevant result. Let $u: \partial D \to \mathbb{R}$ be a continuous function, where $D$ is the open unit disk in the complex plane. What are ...
0
votes
1answer
36 views

Is the domain of a complex function always open? Is $\mathbb C$\ (the domain) always of measure zero? What if the function is holomorphic?

Usually the domain of a complex function is $\mathbb C\backslash\{z\in\mathbb C~:~z \text{ is a singularity of } f \}$ So I guess it must always be $\mathbb C\backslash\{\text{a set of points}\}$. ...
0
votes
1answer
29 views

Is $\mathfrak Rf(a+re^{it})=\mathfrak Rf(a)~~\forall t\in[0,2\pi]$ necessary to prove that $\mathfrak Rf$ is constant?

$\Omega\subset\mathbb C$ is open connected and bounded, $f:\Omega\mapsto\Bbb C$ is holomorphic and continuous on $\overline{\Omega}$ and the real part $\mathfrak {R} f(z)$ attains its maximum at $a\in ...
5
votes
1answer
62 views

Prove that $T$ is a bounded operator on a disk algebra and prove the existence of a Borel measure on a boundary of an open unit disk.

Let $A(D)$ be the space of holomorphic functions on the open unit disk $D$ and continuous on the closed disk $\bar{D}$. Then $A(D)$ is a Banach space if we set $\|f\|=\sup\{|f(z)|:z\in\bar{D}\}$. For $...
1
vote
0answers
30 views

Is this statement sufficient?

Let consider $F:U \to \mathbb{C}, z \mapsto \int \limits_I f(z,t)\mathrm{d}t$ where $U$ is an open set of $\mathbb{C}$ and $I$ an interval of $\mathbb{R}$. Here is the statement : If for all $z \...
0
votes
1answer
30 views

Find $\lim _ { z \rightarrow z _ { 0 } } \frac { f ( z ) } { g ( z ) }$ with f and g holomorphic and $z_0$ a zero of $f$ and $g$

Problem : Let $f$ and $g$ holomorphic within the neighbourhood of $z_0$. Knowing that $z_0$ is a zero of order $k$ of $f$, and a zero of order $l$ of $g$ with $l>k$ My Answer : Since f is ...
0
votes
1answer
23 views

Let $f (z) = u+iv$ be an analytic function, then show that $ (∂^2/∂x^2 + ∂^2/∂y^2)|f(z)|^2 = 4|f'(z)|^2 $.

Let $f (z) = u+iv$ be an analytic function, then show that $$(∂^2/∂x^2 + ∂^2/∂y^2)|f(z)|^2 = 4|f'(z)|^2\,.$$ $f(z) = u + iv $ $ϕ = |f(z)|^2 = u^2 + v^2 $ $f'(z) = ∂u/∂x + i∂v/∂x $ $|f'(z)|^2 = (∂u/...
0
votes
0answers
20 views

Holomorphic function with zero imaginary part over the unit circle is constant [duplicate]

I have this problem I don't know how to approach: let $f$ be continuous on the closed unit disk of the complex plane $\bar{D}(0,1)$ and holomorphic on its interior $D(0,1)$ s.t. its imaginary part is ...
0
votes
1answer
33 views

Proving a certain holomorphic function is polinomic

Suppose we have an holomorphic function $f$ on the open unit disk $D(0,1)$ s.t.: $$\forall r\in (0,1) \exists n\in \mathbb{N}| \max_{C(0,r)}|f|=r^n$$ prove that $f$ is polinomic. Honestly I don't ...
2
votes
2answers
204 views

Function holomorphic in the units disk with different bound

Suppose $f$ is continuous in the closed unit disk $\bar{D}(0,1)$ and holomorphic over its interior $D(0,1)$. Moreover suppose that for $|z|=1$ we have: $\Re(z)\leq0\Rightarrow |f(z)|\leq\ 1$ $\Re(z)&...
0
votes
1answer
35 views

Complex Analysis on Holomorphic Anti-derivatives

Question: Let $U_{1} \subseteq U_2 \subseteq U_{3} \subseteq ... \subseteq \mathbb{C}$ be connected open sets and let $U = \cup_{i = 1}^{\infty} U_i$. Let $f$ be holomorphic on $U$. Suppose for each $...
0
votes
2answers
26 views

Sequence of entire function that converges uniformly over on sets with empty interior

I have to prove that the sequence of entire functions: $$f_n(z)=\frac 1n \sin(nz)$$ converges uniformly over $\mathbb{R}$ (and this I managed to verify) but doesn't on every set with non-empty ...
0
votes
1answer
41 views

Identity theorem for a holomorphic funtion defined near zero

I have to show, whether there is a holomorphic funtion $f$ defined in an open neighborhood of zero, such that: $$ f\left(\frac{1}{n}\right)=(-1)^n \frac{1}{n^3}$$ for all positive integer $n$. My ...
0
votes
1answer
23 views

Entire function invariant by translation is constant [duplicate]

I need to apply Liouville theorem ("entire bounded complex functions are constant") to prove that an entire function satisfying: $$f(z)=f(z+1)=f(z+i)$$ for all complex numbers $z$ is constant. I'm ...
3
votes
1answer
68 views

Holomorphic function on an annulus

Let $f$ be a holomorphic function on the set $U=\{z \in \mathbb{C}: 1 \leq |z| \leq \pi \}$. Assume that $max_{|z|=1}|f(z)| \leq 1$ and $max_{|z|=\pi}|f(z)| \leq \pi^{\pi}$. How to prove that $...
0
votes
1answer
18 views

Is there a biholomorphic map between a simply connected domain and non simply connected domain?

Is there a biholomorphic map between a simply connected domain and non simply connected domain? I am not sure how to approach this question. This is not a homework question but one I simply came ...
0
votes
1answer
44 views

Example of function $f\in C^{\infty}$ but not holomorphic

I know that holomorphic function are infinitely differentiable . I think converse not true .I am searching for counterexample. But I did not get . Please can anyone suggest me how to find such ...
0
votes
1answer
23 views

Harmonic holomorphic function in Ω

$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$ $$v(x,y) = \int_0^1 (yu{\Tiny x} (tx,ty)-xu{\Tiny y} (tx,ty)) dt$$ Prove that there exists a holomorphic function $u+...
-1
votes
1answer
39 views

Evaluate a complex function

Let $\varepsilon>0$. Let $f:B_{\varepsilon}(0)\rightarrow\mathbb{C} $ be an analytic function such that $f(0)=0$, $f(a)=a$ for some $a\in B_{\varepsilon}(0)-\{0\}$ and $\{0,a\}$ are the only fixed ...
0
votes
1answer
51 views

Find points where $f(z)$ is holomophic and compute $f'(z)$ [closed]

If we have a complex function $f(x+iy) = \cos(x)\cos(y)-i\sin(x)\sin(y)$ for $x,y ∈ {\rm I\!R}$ Find the set of points at which f:$\mathbb{C}$ $\Rightarrow$ $\mathbb{C}$ are holomorphic and compute ...
3
votes
2answers
56 views

If $f$ is holomorphic on a domain $D$, and satisfies $a\operatorname{Re}(f(z))+b\operatorname{Im}(f(z))=c$ for all $z\in D$, then $f$ is constant

The proposition that I am required to prove is the following Let $D \subset \mathbb{C}$ be a domain and suppose $f(x, y)=u(x, y)+iv(x, y)$ is holomorphic on $D$. Let $a, b, c\in \mathbb{R}$ such ...
0
votes
2answers
27 views

Example of holomorphic function with no natural root

If $f$ is holomorphic over a simply connected set $\Omega\subset\Bbb C$ and $f(z)\ne0\ \forall z\in\Omega$ then it is known that for any $n\in\Bbb N^*$ there exists a holomorphic function $g$ such ...
1
vote
1answer
25 views

Holomorphic functions reflected through segments that aren't on the real axis

My question concerns using the Schwarz Reflection principle (or symmetry principle) to reflect regions in the domain of a holomorphic function into a symmetric (with respect to a line segment) region ...
0
votes
1answer
17 views

Find all holomorphic functions that satisfy a condition [duplicate]

Find all holomorphic funtion $f:B(0,1)\mapsto B(1,4)$ s.t. $f(0)=3$ and $f(1/2)=1$ $B(a,r)$ is the open ball with centre a and radius r. I think that maybe Schwarz lemma will help, but dont know how....
1
vote
1answer
25 views

Complex Analytic Function's Power Series

Studying complex function, I'm facing some fundamental question: Given open and bounded set $\Omega \in \mathbb{C}$ and $f:\Omega \rightarrow \Omega$ holomorphic on $\Omega$ which $f(z_{0})=z_{0}$. $f$...
-2
votes
2answers
59 views

Finding the residues of $\frac{\cos z -1}{(e^z-1)^2}$.

I've found the poles of $\frac{\cos z -1}{(e^z-1)^2}$ to be double poles at each $z_k = 2k\pi i$, where $k\in\mathbb{Z}$ and $k\neq 0$. (At $k=0$ this is a removable singularity instead.) I have no ...
3
votes
1answer
69 views

$f$ is an entire function s.t $|f(z)|=1$ $\forall z \in \Bbb R$. prove that $f$ has no zeros in $\Bbb C$

My attempt: I was trying to apply identity theorem that if the zeros of the function do have any limit point and it will be a zero function but setting $g(z)=f(z)-1$ will not help me. Can anyone help ...
8
votes
1answer
90 views

Why is it called a holomorphic function?

Why is it called a Holomorphic function? The "Holo" means "entire" and "morphē" means "form" or "apparence", cf wiki. I understand the "entire", because a holomorphic function is differentiable on the ...
0
votes
1answer
54 views

Show that $\sum_{r=1}^\infty \frac{1}{(r-z)^2}$ is holomorphic on $\mathbb{C}\backslash\mathbb{N}$

I'm asked to show that $\displaystyle\sum_{r=1}^\infty \dfrac{1}{(r-z)^2}$ is holomorphic on $\mathbb{C}\backslash\mathbb{N}$, and am given the hint that for any $z \in \mathbb{C}\backslash\mathbb{N}$ ...
3
votes
1answer
58 views

Is the Lambert W function analytic? If not everywhere then on what set is it analytic?

I would appreciate if someone can help me answer the following questions. Although I read several papers and documents on the Lambert W function, I could not assess on what set is this function (or ...
0
votes
1answer
18 views

Transformation of holomorphic functions

$T: H(G) \rightarrow H(G)$, where $H(G)$ is the set of holomorphic functions and $T(f)=f'$. Then how do we prove that $f'$ is continuous?
3
votes
2answers
39 views

holomorphic functions, roots of unity and harmonic numbers

If $f$ is a non-constant holomorphic function such that, for all $z \in \mathbb{C}$, exists a $c \in \mathbb{C}$ where $f(cz) = f(z),$ then $c$ must be a $n$-th root of unity, or there exists some ...
0
votes
2answers
27 views

Show existence of holomorphic function $h$ such that $e^{h(z)}=1+z^5+z^{10}$

Let $U=D_{1/2}(0)$. Show that there is a holomorphic function $h:U\to\mathbb C$ such that $$e^{h(z)}=1+z^5+z^{10}$$ My proof: Such a function exists if $f(z):=1+z^5+z^{10}$ has a logarithm on $U$, ...
1
vote
1answer
35 views

Why is $ I = \int_\Omega \frac{1}{z-\zeta}\Delta u(\zeta) d\zeta = -2i \int_{\partial \Omega} \frac{1}{z-\zeta}\partial u(\zeta) d\zeta. $?

I am unsure of some notation and also how a particular identity was derived. I read in a paper that because $\Delta = 4 \partial \bar \partial$ we have $$ I = \int_\Omega \frac{1}{z-\zeta}\Delta u(\...
2
votes
3answers
111 views

Existance of an analytic function on unit disc

Is there an analytic function $f:B_1(0)\to B_1(0)$ such that $f(0)=1/2$ and $f^{\prime}(0)=3/4$? If it exists, is it unique? The answer to the first part of the question is affirmative. We can use ...
0
votes
1answer
17 views

Biholomorphic functions and delaunay triangulation

Lets have a look at the two simply connected domains $D,G \subset \mathbb{C}$ and a biholomorphic function $f:D \rightarrow G$ which maps $D$ conformal onto $G$. For some $n \in \mathbb{N}$ there ...
1
vote
1answer
36 views

Use maximum modules principle to prove that if $\sup_{|z| = R}|f(z)|\leq AR^{k} + B$ with $f$ entire, then $f$ is a polynomial

Use Cauchy inequality or maximum modules principle to prove that if $f$ is entire function that satisfies $$\sup_{|z| = R}|f(z)|\leq AR^{k} + B$$ for all $R >0$, for some $k \in \mathbb{Z}$ and ...
1
vote
3answers
54 views

Finding all differentiable $f(z) = u(x) + iv(y)$ in $\mathbb{C}$ where $u(x),v(y)$ are real valued functions.

Finding all differentiable $f(z) = u(x) + iv(y)$ in $\mathbb{C}$ where $u(x),v(y)$ are real valued functions. I’m not sure what to do. Would $f$ be differentiable simply if and only if both $u$ and $...
0
votes
1answer
22 views

Find biholomorphic function with certain property

I'm working on the current problem: Find a biholomorphic function $\varphi: S\to H$ where $S=\{z\in\mathbb C| 0<\Re(z)<1\}$ and $H=\{z\in \mathbb C| \Im(z)>0\}$. Find a biholomorphic ...
1
vote
1answer
33 views

Is $\frac{\sin(2z)}{e^z-1}$ holomorphic at $z=0$?

Is $f(z)=\frac{\sin(2z)}{e^z-1}$ holomorphic at $z=0$? The domain of $f$ is $\mathbb C$\ $\{0\}$.So it's not holomorphic at $0$?
1
vote
1answer
24 views

Extending uniform convergence of analytic functions on larger domains

Let $f_k, f: ]-\infty , 1 [ \to \mathbb {R}$ be analytic functions. Suppose $f_k $ converges uniformly to $f $ on $]-\infty,0] $. Is it true that $f_k$ converges to $f$ on $]-\infty, \epsilon [$ for ...
1
vote
1answer
51 views

Constructing Möbius transformation

In general my approach to construct a Möbius transformation $\varphi$ between two simply connected domains $G_1$ and $G_2$ is to take 3 points on each boundary and map them onto each other. The cross-...
1
vote
0answers
32 views

Harmonic holomorphic function in $Ω$

$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$ $$v(x,y) = \int_0^1 (yu{\Tiny x} (sx,sy)-xu{\Tiny y} (sx,sy)) ds$$ Prove that there exists a holomorphic function $u+...