Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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3answers
27 views

Ordering complex numbers - When is it right and when is it not?

I know there are a lot of questions related to this topic. But, I have one specific doubt. If we order complex numbers, does that mean that we are wrong all the time? If we say $4 + 3i < 5 + 7i$. I ...
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0answers
22 views

Does integrability implies integrability with respect to the harmonic measure?

Let $D$ be a bounded domain in $\mathbb{R}^{d}$ ($d>1$) and $f$ a measurable function on $D$. Suppose $K$ is a compact of the boundary of $ D$ and $\omega_{x}$ designates the harmonic measure of $...
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2answers
33 views

Definitions of Holomorphic

I have seen holomorphic defined two different ways. The first is that a function f is holomorphic at $z$ if the limit $\lim\limits_{h \to 0} \dfrac{f(z + h) - f(z)}{h}$ exists and the second is that ...
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4answers
45 views

What is a positively oriented Jordan curve?

I'm reading the book "Methods of Nonlinear Analysis" written by Pavel Drábek. In this book there's the following proposition: Let $\gamma$ be a positively oriented Jordan curve, $\sigma(B)\subset ...
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0answers
29 views

Modern complex analysis book (any suggestions ?)

Can someone suggest any modern books for complex analysis (in pdf if possible)? With nice looking text and formulas, examples, chapters, chapter reviews and so on. It is easier and more interesting ...
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1answer
22 views

A Proof of Residue Theorem on a Compact Riemann Surface

Usually a proof of the Residue Theorem on a Compact Riemann Surface uses the crucial fact that Holomorphic forms are closed. I tried to write a proof and somehow I didn't use that fact anywhere. Can ...
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4answers
109 views

Is $(-1)^{2.16}$ a real number?

A lot of calculators actually agree with me saying that it is defined and the result equals 1, which makes sense to me because: $$ (-1)^{2.16} = (-1)^2 \cdot (-1)^{0.16} = (-1)^2\cdot\sqrt[100]{(-1)^{...
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1answer
39 views

$0$ in Polar form

I want to prove that for $z_1, z_2 \in \mathbb{C}$, then $$ \text{arg}_I (z_1 z_2) = \text{arg}_I (z_1 )+ \text{arg}_I (z_2) $$ The proof is staighforward, but I was wondering what to do about $z ...
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0answers
22 views

Phragmén-Lindelöf - Extend bounds of the form $(\frac{c}{t})^{-\frac{c}{2}}$ of a real positive function to $\mathbb{C}^+$?

After the great help I got last time, I decided to ask for help again with a problem Im struggling with. Hello people, Let $ K:(0,\infty)\to [0,\infty) $ be a function, that can be extended ...
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1answer
37 views

Plotting $\sqrt{z}$ where z is a complex number

I try to plot the image of the domain $-\infty<x<\infty$ and $0<y<\pi$ under the function $w=\sqrt{z}$. I've got a hint that one boundary is a hyperbola. I tried sth like that: $w^2=z=x+...
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0answers
28 views

A Homotopy-Based Proof of the Cauchy's Integral Formula?

Suppose that we have proved the Cauchy's Integral Formula for positively oriented circles, and that now we want to prove a more general version for piecewise smooth Jordan curves. This can be done by ...
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2answers
31 views

Analyticity of function $g(w)$ given that $g(w)=f(z)$ and $f(z)$ is analytic

Suppose you have a function $f(z)$ which is entire on the complex $z$ plane. Now consider variable $w=z^2$ and define a function $g(w)$ on the $w$ plane such that $g(w)=f(z)$. What if anything can be ...
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1answer
21 views

Real-valued bounded analytic functions on the unit disc

Let $f: \overline{\mathbb{D}} \to \mathbb{R}^+$ be a real (positive) valued function on the closed unit disc that is bounded and analytic on $\mathbb{D}$ (open unit disc) and $$\lim_{|z| \to 1}f(z) = ...
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1answer
79 views

Fourier transform of $f(x)=\frac{4}{3+2x+x^2}$

I want to find the Fourier transform of $$f(x)=\frac{4}{3+2x+x^2}$$ We worked with the following definition: $$\hat f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-itx}dt$$ Now, $f(x)=\...
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1answer
57 views

Prove that if analytic function $f$ is such $f(0) \neq 0$, it has no zeroes in a certain disk [duplicate]

Problem. Assume $\displaystyle f(z)=\sum_{n=0}^\infty a_n z^n$ is analytic in $\overline{U}=\{|z|\leqslant R\}$ and $a_0\ne 0$。Prove: $f$ has no zeroes in the circular disk $\left \{|z|< \dfrac{|...
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0answers
16 views

Explicit holomorphic differentials on $y^3 = P(z)$ riemann surface

Let X be a compact Riemann surface of genus $g$ corresponding to the equation $w^3 = P(z)$, P is a polynomial without multiple roots and $degP = 8$ It's known that there are $g$ holomorphic ...
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2answers
40 views

Why do we have to specifically define a branch of the complex square root function?

If $z=re^{i\theta}$, then $\sqrt{z}=\sqrt{r}e^{i\frac{\theta}{2}}$. Isn't this a well defined function on the whole complex plane? Why do we need to define this as the function $\mathbb C \setminus \...
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1answer
29 views

Positive real part of derivative imply injectivity?

Suppose that $U\subset\mathbb{C}$ is open, connected, and that the derivative of the analytic functions $f:U\to\mathbb{C}$ has strictly positive real part. Does it follow that $f$ is injective on $U$?...
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0answers
13 views

Finding meromorphic functions with maximal value in a preset point

I am having difficulties with finding meromorphic functions that have a maximum value in a point I am given in advance. For example, I want to find a function, meromorphic on a unit disk, that reaches ...
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1answer
34 views

Montel Great theorem

Montel's Great theorem states Let $\mathcal{F}$ be a collection of analytic functions on a region $\Omega$ such that all of the $f\in \mathcal{F}$ omit the same two values $a,b$. Then the family is ...
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0answers
17 views

Explicit Example Dense Orbit Composition Operator

Its relatively straightforward to show that the composition operator $$ C_{\phi}:f\mapsto f\circ \phi, $$ is topologically transitive on $H(\mathbb{C})$, the space of holomorphic functions on $\...
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1answer
34 views

What are some resources for learning mathematical writing?

I'm looking for recommendations for some resources which will teach me how to convey ideas through consistent mathematical writing? For example, the book mathematical thinking and writing is one ...
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0answers
29 views

An application of the Paley-Wiener theorem

Extract of an article: "the Laplace transform of $T(t)f$ is an entire function and since the resolvent is meromorphic of finite exponential type it must be of finite exponential type, say $\nu$, too. ...
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1answer
44 views

Does this system of infinite equations has an (almost) unique solution?

Let $a_1,\dots ,a_n \in \Bbb C$, consider the following system of equations $$\begin{cases} x_1+ \cdots+ x_n=a_1 \\ {x_1}^2+\cdots+{x_n}^2=a_2 \\ \qquad \qquad \vdots \\ {x_1}^n+\cdots+{x_n}^n=a_n \...
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1answer
25 views

the question about the proof of analyticity of gamma function

On the clip , https://youtu.be/E7NNc-AM7vQ?t=385 (at the current time), the speaker checkes that $P_n(z)$ = $\int_{n}^{n+1} t^{z-1}e^{-t}dt$ is analytic function. To do so, he referred "Use Thm5 (...
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2answers
26 views

The coefficients of $\sum_{k=0}^na_{n,k}\tan^k z$ are zero if $n,k$ are both even or both odd

There is a formula for the $n$-th derivative of tangens, given by $$\tan^{(n)}(z)=\sum_{k=0}^{n+1}a_{n,k}\tan^k z$$ for $n\in \{0,1,2,\dots\}$, where $a_{n,k}$ are non-negative integers. The problem I ...
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3answers
33 views

derivative of complex composite functions

I want to derivate a real-valued function of real variable, defined as: $$L(x) = f(g(x)), $$ where $g$ is a complex-valued function of real variable and $f$ a real-valued function of complex variable....
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1answer
60 views

A question about the proof of fundamental theorem of algebra

I'm reading the proof of fundamental theorem of algebra from textbook Analysis I by Amann. I have a problem understanding the part: Hence we can write $q$ in the form $$q=1+\alpha X^{k}+...
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0answers
30 views

Describe the Riemann Surface for $z^2 = w + \frac{1}{w}$

Describe the Riemann Surface for $z^2 = w + \frac{1}{w}$. Not sure if my process is correct. We can see $z = \frac{\sqrt{w^2+1}}{\sqrt{w}}$. So there are branch points at 0, i, -i. We have two layers, ...
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1answer
74 views

identities other than $(a-b)^2=(a+b)^2-4ab$? [on hold]

We know that for any integer (or real in general) the following equation $$ (a-b)^2=(a+b)^2-4ab $$ holds for all $a,b$. I am looking for some other identities which involves $a+b$ and $ab$ and ...
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0answers
41 views

How to find the branch cuts of $\sqrt{g(z)}$ and the contour integral $\int_{z_1}^{z_2}d z\sqrt{g(z)}$

I need to evaluate the following integral: \begin{equation} \int_{z_1}^{z_2} d z \sqrt{g(z)}, \end{equation} where the function $g(z)$ is given by \begin{equation} g(z)=-\left(\alpha-\frac{\beta}{...
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1answer
47 views

In showing the principal branch of $\operatorname{Log}(z)$ is holomorphic on $\Bbb C\smallsetminus \mathbb{R}^{\leq 0}$, where is the branch used?

Suppose we wanted to show that the principal branch - from here on denoted by $\operatorname{Log}(z)$- of the complex logarithm is holomorphic on its domain $\mathbb{C} \smallsetminus \mathbb{R}^{\leq ...
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1answer
33 views

rewrite function with removable singularity

Suppose the function $f(z)$ has a removable singularity at $z_o$. Why can I rewrite this function as $f(z)=(z-z_0)^kg(z)$ for some $k\in \mathbb{N}$ and a holomorphic function $g(z)$ where $g(z_0)\...
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0answers
7 views

Time frequency reassignment for complex signal

Given the following, $$f(l)=\sum_{k}f_{k}(l)e^{-i\phi_{k}l}$$ $$W_{f}(\tau,v)=\int_{\mathbb{R}}f(l+\tau/2)f^{*}(l-\tau/2)e^{-ivl}\partial l$$ $$W_{g}(\tau-t,v-\omega)=\int_{\mathbb{R}}g(l+(\tau-t)/2)...
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0answers
12 views

Bergman spaces and Dirichlet spaces equvalency

Suppose $\alpha > -1$ and $p>0$. $A_{\alpha}^{p}$ is Bergman space and $D_{\alpha}^{p}$ is Dirchlet space. I want to know exact proof of why $$D_{\alpha}^{p}= A_{\alpha-p}^{p}$$ for $\alpha> ...
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1answer
24 views

which domain does this Möbius map to $\Re(w) > 0$?

Given the following Möbius: $$ w = T(z) = \frac{1+z}{1-z} $$ How could I find the domain of $Z$ which $T$ maps to $\{\Re(w)>0\} $? I tried to inverse $T$ and got: $$ z = T^{-1}=\frac{w-1}{w+1} $$ ...
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2answers
26 views

An analytic function in a compact region has finitely many zeros

I’m trying to solve the following problem, but I can’t. I need your help. Recall (Sec. 11) that a point z is an accumulation point of a set S if each deleted neighborhood of z contains at least one ...
2
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0answers
32 views

Finding the branch of a function

I am having a really hard time understanding exactly how to determine the Riemann surface of a complex function f(z). I understand the concept: images of these complex functions are periodic, and so ...
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1answer
65 views

How to solve a complex exponential functional equation which contains the multivalued argument function of complex numbers

I am trying to solve a complex exponential functional equation. A real to complex function $\gamma(t):\mathbb{R}\to\mathbb{C}$ has the following form \begin{align} \gamma(t)=e^{rt}\cdot e^{i(H(t)+2k\...
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0answers
70 views

Complex analysis using Kronecker delta..

I have some trouble with proving the formula below; $$\sum_{k=1}^n \sin\Big(\frac{kr \pi}{n+1}\Big)\sin\Big(\frac{ks \pi}{n+1}\Big) = \frac{n+1}{2}\delta_{rs}$$ by making use of complex analysis for ...
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1answer
24 views

proving radius of convergence for power series

Let $\sum a_n z^n$ be a power series. Proof that if $lim _ {n \rightarrow \infty} |\dfrac{a_{n+1}}{a_n}|$ exists, then it is equal to $\dfrac{1}{R}$, where $R$ is the radius of convergence. So, using ...
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1answer
21 views

Continuity of partials of real and imaginary parts of holomorphic function

I have repeatedly come across the following sort of argument: Let $f = u+iv$ be a holomorphic function. Then since it is infinitely differentiable, so are $u$ and $v$. Thus, the partial ...
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0answers
12 views

Concerning local development in two variables

In one complex variable, a meromorphic function $f$ with a simple pole at $s=s_0$ can be written in the form $$f(s) = \frac{A}{s-s_0} + H(s),$$ where $A$ is a constant (the residue at $s=s_0$) and $H$...
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0answers
48 views

(Branch cut of z^2) Can someone explain this picture?

I am trying to understand exactly what is going on in the picture below: From what I understand so far, these are two complex planes. The left one is z, and the right one is the image of z under $f(z)...
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1answer
35 views

If $f$ is a holomorphic function then $|f|$ is strictly sub-harmonic.

I got this problem in an exam and it looks so simple. Of course the cauchy’s formula for a holomorphic function $f$ allows one to say that $|f|$ is sub harmonic. But I am not able to see why it should ...
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2answers
29 views

Show that $\frac{1}{2}(|z|+|w|)|\frac{z}{|z|} + \frac{w}{|w|}| \le |z+w|$ with $z \neq 0$ and $w\neq0$ [duplicate]

Let $z,w \in \mathbb{C}$ Show that $\frac{1}{2}(|z|+|w|)|\frac{z}{|z|} + \frac{w}{|w|}| \le |z+w|$ with $z \neq 0$ and $w\neq0$ I know that $|z+w|\le |z| + |w|$ And that $z \cdot\bar z = |z|^2$ I ...
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0answers
27 views

Intuition behind monodromy group.

What is a monodromy group from an intuitive standpoint with respect to the roots of a polynomial, or even branch points of a complex function? I realize this question was asked in a slightly different ...
1
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1answer
25 views

Identity theorem and function $f(z) = \sin{\frac{\pi}{z-1}}$ on unit disc

I am having problems understanding identity theorem (wikipedia) in complex analysis. I have a holomorphic function $f(z) = \sin{\frac{\pi}{z-1}}$ defined on the unit disc except for the $1$. Roots of ...
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0answers
27 views

Small doubt in the proof of Hurwitz theorem

I was going through the proof of Hurwitz theorem from T. Gamelin.At one point author said sequence $\frac{f_k'}{f_k}\to\frac{f'}{f}$ converges uniformly on$\mid z-z_0\mid=\rho$.($f_k $ converges ...
0
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1answer
21 views

Definition of meromorphic funciton on complex manifold

I'm having trouble finding a definition for a meromorphic function from the Riemann sphere to itself. Denoting the sphere $\hat{\mathbb{C}}$ we have that $$f:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}...