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.

727 questions
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\mathbb C\backslash\{\text{a set of points}\}$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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 $f(z_{0})=z_{0}$. $f$...
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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 ...
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$f$ 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 ...
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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 ...
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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 $z \in \mathbb{C}\backslash\mathbb{N}$ ...
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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 ...
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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?
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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 ...
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 $U$, ...