The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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Which permutations of $\mathbb{C}$ commute with the Riemann zeta function?

I'm trying to figure out whether the permutations of $\mathbb{C}$ which commute with the Riemann $\zeta$ function are necessarily continuous or not. Obviously both the identity and the complex ...
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1answer
38 views

Complex Roots Made Visible [closed]

Can anyone point me to a pdf of the article, "Complex Roots Made Visible", by Norton and Lotto. Thanks, Ron
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1answer
34 views

Conformal mapping and its application in finding roots of polynomial

So for a polynomial, if we want to find the roots in a complex plane. Rouche's theorem is the first tool in my head. However, I saw several problems of finding the roots in the first quadrant or upper ...
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3answers
55 views

Using complex variables to find sums of Fourier series

Use complex variables to find the sum of the Fourier Series: $$\sin(\theta) + \frac{\sin(2\theta)}{2^{2}} + \frac{\sin(3\theta)}{2^{3}}+\cdots$$ where $\theta$ is a real variable.
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1answer
49 views

Green's function for Helmholtz equation for the plane with a hole

That is find $G$ which satisfies \begin{align} (\nabla^2+k^2)G(\mathbf{x}, \mathbf{y},\omega) = \delta(\mathbf{x}- \mathbf{y}) \end{align} subject to $$\frac{\partial G}{\partial y_n} = 0 ...
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1answer
74 views

Identity Principle type question: Prove that $f=g$

While reading a complex analysis textbook the following assertion came up Since $f,g:D\equiv D(a,r) \to \mathbb{C}$ are analytic and injective functions such that $f(D)=g(D)$, $f(a)=g(a)$ and ...
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40 views

Analytic $N$th roots: can my statements here be generalized?

I'm generally having trouble seeing when and why analytic functions have analytic $N$th roots. I know the following statements to be true, but I know that they can also be generalized in various ...
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1answer
53 views

Let $f(z)$ be a function analytic in a domain containing the segment $[0,1]$ and satisfying $f(z+1)=azf(z)+p(z)$.

Let $f(z)$ be a function analytic in a domain containing the segment $[0,1]$ and satisfying $$ f(z+1)=azf(z)+p(z) $$ in that domain, where $a\in\mathbb{R}$ and $p$ is a polynomial. Show that ...
5
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1answer
257 views

Riemann hypothesis reformulation - again

Yesterday I started to write a paper about the reformulation of the Riemann Hypothesis. My idea was to map the function such that all of the trivial zeros are outside of the unit disk, and the ...
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1answer
44 views

Show that Mergelyan's theorem cannot extend to the case in which $S^2-K$ has infinitely many components.

This is an exercise in W. Rudin's Real and Complex Analysis. For $n=1,2,\ldots$, let $D_n=D(\alpha_n;r_n)$ be disjoint open discs in (the unit open disk) $U$ whose union $V$ is dense in $U$, ...
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1answer
24 views

When does $-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$ where $z$ is a complex variable?

Let $z$ be a complex variable. Is there someone who can show me when does :$$-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$$ Note: I have tried using trigonometric formulas but it didn't work. Maybe I ...
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1answer
82 views

Equivalence of holomorphic functions

Given that $$\left(1-\frac{z}{\zeta_j}\right)^{-z}=\sum\limits_{k=1}^\chi\frac{z^k}{k\zeta_j^k},$$ where $\chi$ is the largest nonnegative integer $k$ for which ...
2
votes
1answer
54 views

Do analytic functions on open subsets of $\mathbb{C}$ with an analytic square root form a sheaf? [duplicate]

I'm trying to learn algebraic geometry and am trying to think about what kinds of things are presheafs but not sheafs. One exercise I had was to show that bounded holomorphic functions on open ...
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2answers
35 views

Complex Analysis ( Open/Closed Set).

let $z = re^{i\theta}$ , How do we prove that , $0\leq \operatorname{arg}(z)\leq\dfrac{\pi}{4}$ ($z \neq 0$), is neither a open set nor a closed set. $\operatorname{arg}(z)$ is nothing but $\theta$ ...
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3answers
45 views

Is the boundedness necessary to extend harmonically?

"If $u$ is harmonic and bounded in the punctured disk $0<|z|< \rho$, then $u$ can be extended harmonically to the disk $|z|<\rho$ harmonically." This fact has been shown here. My Question ...
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38 views

Complex dot product

I know that the complex dot product is defined as $\boldsymbol{a}\cdot\boldsymbol{b}=\sum_{i}a_ib_i^*$. Is there a standard name for the operator ...
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1answer
68 views

How to differntiate $\int_{0}^{2\pi} u(re^{i\theta}) d\theta$?

Suppose $u$ is a twice continuously differentiable function on $a< |z|<b, \ z\in \mathbb C,$ which is harmonic that is, it satisfies $u_{rr}+\frac{1}{r}u_r + u_{\theta \theta}=0.$ (If we put ...
2
votes
1answer
51 views

integrals of exponential functions over the real axis

How to evaluate the integral $$ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})dx? $$ I intend to change the variable $x$ to $\tan t$ but failed... How to solve it?
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2answers
37 views

A question regarding a proof in Ahlfors

Ahlfors says the following: if $f (z) $ is analytic on a disc, then its integral along any closed path contained in the disc is $0$. The proof for this is the following: Let $F (z)=\int_\sigma {f ...
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1answer
27 views

If $\phi$ is entire and satisfies $|\phi(z)| \leq e^{|z|}$, then $|\phi'(z)| \leq c e^{|z|}$ for some $c > 0$.

If $\phi$ is entire and satisfies $|\phi(z)| \leq e^{|z|}$, then $|\phi'(z)| \leq c e^{|z|}$ for some $c > 0$. I saw this problem on a practice qual but I had no idea what to do. It looks ...
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1answer
49 views

Functions with real domain but complex range, do they have any use?

For example if we define the square root function like this: $$\text{Sqrt}({x})= \begin{cases} \sqrt{x} & x\geq 0 \\ i\sqrt{-x} & x<0 \end{cases}$$ Or we could have an exponential ...
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2answers
53 views

Show that $\lim\limits_{z \to a}\frac{\log|f(z) - f(a)|}{\log |z - a|}$ is an integer.

Let $f$ be analytic in a neighborhood of $a$. Show that $\lim\limits_{z \to a}\frac{\log|f(z) - f(a)|}{\log |z - a|}$ exists and is an integer. We have $$\frac{\log|f(z) - f(a)|}{\log |z - a|} = ...
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1answer
59 views

Simple example of the use of sheaves

What would be (one of) the simplest example of a mathematical result which is solved using the concept of sheaves?
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35 views

Holomorphic proper function on $\mathbb{C}$ is a polynomial

I want to show that every holomorphic proper map $f:\mathbb{C}\to \mathbb{C}$ is a polynomial. Since $f$ is continous and proper, it can be extended to a continous map $f:S^2\to S^2$, where $S^2$ is ...
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1answer
36 views

Prove that a harmonic function is an open map.

I'm trying to solve the following exercise of the book Functions of one complex variable, John B. Conway on page 255: 4. Prove that a harmonic function is an open map. (Hint: Use the fact that the ...
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1answer
40 views

Requirement for a given function to be smooth

I have quite a basic question about the derivatives. My uncertainty comes mainly from the fact that I don't know how the complex logarithm behaves. Here is the description (this task is not ...
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1answer
35 views

Uniqueness of harmonic function with Mixed Dirichlet Neumann condition

Let $u \colon \{\mbox{Im } z>0\}\subset\mathbb{C}\to \mathbb{R}$ be a positive harmonic function in the upper half plane, i.e $$ \Delta u=0,\,\, \mbox{for}\,\mbox{ Im } z>0. $$ Consider now the ...
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2answers
34 views

A doubt regarding a property of a complex function

I just wanted to confirm that when we define a complex function $f (z) $, is it just a function in terms of $z $, or also that in terms of $\overline {z} $? This is because if the latter is true, ...
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1answer
50 views

Variant of Riemann mapping theorem

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk) and let $\Omega$ be non empty open simply connected in $\mathbb C$ and $\Omega \neq \mathbb C.$ Then Riemann mapping theorem tells us that there exists ...
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17 views

Complex Normal Gaussian noise

I would like to create complex normal Gaussian noise with dimensions $(M,N)$ The noise should have zero mean and $var=1$. How can I do so?
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50 views

integral from gradshteyn and ryzhik

I'm interested in evaluating the integral $$ \int_{a}^\infty e^{-x\cosh\alpha}\,K_{\nu}(x\sinh\alpha)\,\frac{dx}{x}, $$ where $a>0$ and $\nu$ is purely imaginary. Here $K$ denotes the MacDonald ...
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0answers
18 views

Analyticity of the outer function of an analytic composition

Let $\mathscr{U}$ be an open neighborhood of the origin of $\mathbb{C}$ and let $F(t,x)$ be a function that is continuous on $\mathbb{C} \times \mathscr{U}$ and that is holomorphic in $\mathscr{U}$ ...
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1answer
20 views

$ \exists f:\mathbb C \setminus D \to \mathbb C$ is bounded one-one holomorphic, how?

We note that there cannot exist bounded one-one holomorphic map $f:\mathbb C \setminus \{0\} \to \mathbb C.$ Put $D=\{z\in \mathbb C: |z|\leq1\}$ (closed disk). My Question: How to show there ...
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1answer
21 views

real coordinates of a complex manifold

I have a naive question about real coordinates of a complex manifold. Let's consider 1-dimensional case for simplicity. Let $X$ be a Riemann surface and $z$ be a local complex coordinate. Then one ...
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1answer
117 views

Show that $f(z)\not=0, \forall z\in \mathbb C$.

Suppose , an entire function maps the real line onto the circle $C=\{z:|z|=R\}, R>0$. Show that $f(z)\not=0, \forall z\in \mathbb C$. I thought through contradictory way but I could not think ...
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1answer
30 views

If $\lim_{|z|\to 1^-}u(z)=0$ then $u\equiv 0$

Let , $u(z)$ be a complex valued harmonic function in $|z|<1$ and $\lim_{|z|\to 1^-}u(z)=0$. Then show that $u(z)$ is identically zero in $|z|<1$. I am unable to understand that from where ...
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1answer
42 views

A question about current and Dirac measure

$0$ can be seen as a divisor of $\mathbb{C}$, and the current $[0]$ is defined as $[0](\varphi)=\varphi(0)$. Why is this reasonable?
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7 views

Characterization of Multivariate Polynomials with Unique Critical Point

I would like information about $\{f\in \mathbb{C}[x_1,\ldots,x_n]:Z(\nabla(f))=\{0\}\}$. Above, the $Z$ denotes the vanishing locus of a function, i.e. the set of points where it vanishes, and ...
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1answer
38 views

Complex Analysis Dense Set Problem

The Problem: Suppose $f(z) = e^{i\theta}z$. Show that if $\theta$ is not a rational multiple of $\pi$, then the orbit of $ z \in \mathbb{C}$ is dense in the circle with radius $|z|$ and at the center ...
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26 views

Proof of Schwarz-Pick Theorem

This question has probably already been asked before, but since I can't find it here (it's probably labeled "Application of Schwarz lemma" among 30 others) I'll repeat it. Let $f: D \rightarrow D$ ...
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0answers
28 views

Proof of Sokhotski-Plemelj theorem

Sokhotski-Plemelj theorem states $$ \phi_i(z)=\frac{1}{2\pi i}\mathcal{P}\int_C\frac{\varphi(\zeta) \,d\zeta}{\zeta-z}+\frac{1}{2}\varphi(z), \, \\ \phi_e(z)=\frac{1}{2\pi ...
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Show that a conformal map of an open disk onto any open disk is necessarily bilinear.

Show that a conformal map of an open disk onto an open disk is necessarily bilinear. Please help me with this proof. Any help will be appreciated. To clarify this, it's not just a map from a open ...
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1answer
64 views

Does there exists bounded one-one holomorphic map $f:\mathbb C \setminus \{ 0 \} \to \mathbb C$? [closed]

(1)Does there exists bounded one-one holomorphic map $f:\mathbb C \setminus \{ 0 \} \to \mathbb C$? (2)Let $X$ be a closed connected subset of $\mathbb C$ and which has more than one element. Does ...
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2answers
21 views

Quotient of space and a group of maps, Riemann surfaces

I've been attempting to study Riemann surfaces, and I have continuously run into this notion which eludes me. I see people write things like $ \mathbb H / <z\mapsto z+1>$ or $\mathbb D / PSL$. I ...
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3answers
23 views

Primitive of a meromorphic function

I found this statement that I cannot justify due to my lack of knowledge in complex analysis (this is not my field of study). Let $D\subset \mathbb{C}$ be the open unitary disc centered at $0$, let ...
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3answers
49 views

Why if a function is holomorphic and injective in neighbourhood of $x_0$ then $f'(x)\ne 0$ in neighbourhood of $x_0$?

Why if a function is complex differentiable and injective in some neighborhood of $x_0$ then its derivative is non zero in that neighborhood? I just don't see how why it is like that. Obviously in ...
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2answers
73 views

Evaluate $\frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta$ [duplicate]

Let $0 < r < 1$. Compute $$\frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta$$ The hint is rewrite this integral as a complex line, but I still don't know how to to ...
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1answer
15 views

Is the following property of analytic functions true?

Let $f $ be an analytic function on $\Bbb {C} $. At any point $z_0$, for every $\epsilon>0$, does there exist a neighbourhood $B (z_0, \delta) $ such that for every $z\in B (z_0, \delta) $, we ...
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1answer
37 views

Value of $\int_C\frac{e^z}{z}dz$ with $C$ unit circle

Compute the integral $$\int_C\frac{e^z}{z}dz$$ where $C$ denotes the unit circle with positive orientation. I was thinking that let $z = e^{it}$, $dz = ie^{it}$, then the integral will become ...
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1answer
46 views

Does there exists biholomorphic map(with suitable condition) from domain to open disk?

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk). We define $f:D\to D$ as $f(z)=z,$ for all $z\in D,$ which is clearly biholomorphic. My Question is: (1) For any $z_0\in D,$ can we choose a ...