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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How to prove that a complex function is differentiable at a point and where is it analytic in its domain

I know the definition of complex differentiable functions and I know the Cauchy-Reimann equations that are used to prove that a function is differentiable in general, but my question is how to prove ...
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7 views

Describing the image of the upper half disk under log z.

i'm not entirely sure how to go about this. Having the function $log(z)$, $z$ can be represented in polar form $z = re^{\ i\ \theta} $. So $log(z) = log(r) + i \theta $. Since i'm asked to map the ...
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28 views

Mistake in this definition from Conway's complex analysis book

I'm reading Conway's complex analysis book and on page 64 the author has enunciated the following definition: However, on page 81 the author has stated that: I think Conway was mistaken in the ...
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7 views

Help with understanding when $\operatorname{Log}(z^k)=k\operatorname(z)$ as well as drawing the function.

For the question I'm dealing with the property $\operatorname(z^k)=k\operatorname(z)$ in which I have to find the largest open set that this property is true when $k$ is a positive integer. I ...
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8 views

How to do calculus with split-complex (hyperbolic) numbers?

TL;DR: how do I define a "split-holomorphic" function? As far as I've heard, there is a notion of split-complex numbers: $z = x+uy$, with $u\not\in \Bbb R $ and $u^2 = 1$. One defines the conjugate ...
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30 views

residue at an essential singularity and the value of the integral

the way i calculated was to open the cosine series and then multiply by $z^7$.by doing this i got the value to be $I=1/4!$ .is it correct?please explain
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2answers
27 views

Proving that rotation is an isometry in the complex plane

Consider the rotation $ρ_θ : \Bbb C → \Bbb C$ about the origin with angle $θ$ in counterclockwise direction; this can be described by the map $ρ_θ(z) = e ^{iθ} z$. Prove that $ρ_θ$ is an isometry of ...
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1answer
14 views

Showing that a function is an isometry of the complex plane and showing that a composition of functions in the complex plane is a translation

a). Let $a ∈ \Bbb{C}$ be fixed. Show that the map $T_a : \Bbb C → \Bbb C$ given by $T_a(z) = z + a$ is an isometry of $\Bbb C$. This is a translation of the complex plane $\Bbb C$. For this first ...
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14 views

A question concerning complex conjugates of constants

If $τ : \Bbb C → \Bbb C$ is given by $τ (z) = e^{iθ} \bar z + a$ then $τ^{−1}(z) = e^{−iθ} \overline {(z − a)}$ for some fixed $a ∈ \Bbb C$. I know that I need to show that $τ τ^{−1} (z) = τ^{−1} τ ...
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5answers
92 views

Give examples showing why $0\cdot \infty$, $\infty/\infty$, and $0/0$ are meaningless

Assuming arithmetic operations on $\overline{\mathbb{C}}$ (that's the extended complex plane) are defined via arithmetic operations on the corresponding sequences, I need to give examples showing why ...
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1answer
13 views

Image of Upper Half Disc under $w = 1/z$

I need to find the image of the upper half disc $|z|<1$, $Im\, z >0$ under the inverse transformation $w = 1/z$. Now, since $|z|<1$, $|z|^{2}<1$. Rewriting this as $z\overline{z}<1$, ...
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0answers
20 views

Functional equation for Hurwitz Zeta

I have been studying Reimann's functional equation (symmetric form) for Zeta(s) and wondering if a similar functional equation has been derived for the Hurwitz Zeta function?
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1answer
14 views

Hybrid equivalence of Polynomial-like maps

I am reading Douady and Hubbards "On the dynamics of polynomial-like mappings". I am relatively new dynamics of complex maps, and I would appreciate some help with aspects of the following. ...
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1answer
18 views

Complex integral over line, similarity with conservative field

Have $\int_C(4z^2-2iz)dz$ integral. Does it depend on choice of path? Tried to express $f(z)=(4z^2-2iz)$, then $f(x+yi)=(4x^2-4y^2+2y)+i(8xy-2x)$ Then $\frac{\delta P}{\delta y}=-8y+2$ And ...
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0answers
17 views

Gaussian integral over complex values

I am trying to evaluate $|\int_{\gamma}e^{z^2}dz|$ for $\gamma$ is a segment from $(-3-3i)$ to $3i$, and fail badly. Tried to express $e^{z^2}$ as Taylor serie, but apparently it got me nowhere. Any ...
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0answers
18 views

Do we have for $\lim_{r\to \infty}\frac{1}{rlog(r)}(\int_0^{2\pi}u(re^{i\theta})d\theta)$ exists for $u$ subharmonic?

Let $u:\mathbb{C}\to \mathbb{R}$ be a subharmonic function. Do we have that the limit $\lim_{r\to \infty}\frac{1}{rlog(r)}(\int_0^{2\pi}u(re^{i\theta})d\theta)$ converges to a (possibly infinite) ...
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1answer
83 views

Can the real part of an entire function be bounded above by a polynomial?

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function such that $Re(f)\le |p(z)|$ for some polynomial, can we derive that $f(z)$ is a polynomial. If $p(z)$ is constant, then this can be shown by ...
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30 views

Verifying formulas for reflection, rotation, and translation in the Complex plane.

1) If $T_a : \Bbb C → \Bbb C$ is given by $T_a(z) = z + a$ then $T^{−1} _a (z) = z − a = T_{−a}(z)$ for some fixed $a ∈ \Bbb C$. 2) If $ρ_θ : \Bbb C → \Bbb C$ is given by $ρ_θ(z) = e^{iθ}z + a$ then ...
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1answer
44 views

When will equality holds in reverse triangle inequality?

Prove the reverse triangle inequality :$|z\pm w|\ge||z|-|w||$ for all $z, w \in \mathbb C$, with equality holds if and only if either $z$ or $w$ is a real multiple of another. I have proved the ...
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0answers
13 views

continuity in complex value function [duplicate]

I want to prove a function $f(t)$ is continuous at $ t_o$. By def of continuity, I need to show that for $\epsilon >0 $, there exist $ \delta>0$ s.t $ |t-t_o| < \delta$ implies $ ...
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1answer
26 views

Having trouble understanding(The proof of); $A \subset \mathbb{C}$ is finite implies the limit point set of $A$ is empty.

So I am self-studying Complex Analysis. The idea behind the proof is clear, and I understand the intuition etc. There are a few statements that are really giving me trouble accepting this proof ...
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1answer
41 views

Proving a fact about complex numbers [duplicate]

How to prove the following: For any complex numbers $z_1,...,z_n$, there exists a subset $E$ of $\{1,...,n\}$ s.t. $$\left| {\sum\limits_{j \in E} {{z_j}} } \right| \geqslant ...
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1answer
27 views

Epsilon delta proofs of theorems of continuity

Can anyone suggest a book which contains epsilon delta prooves for properties and theorems of continuity rather than sequential proofs.
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3answers
50 views

How many entire functions are equal to $\frac{1}{z}$ for $|z| > 1$?

Let $f:\{z\in\mathbb{C}:|z|>1\}\rightarrow\mathbb{C},f(z)=\frac{1}{z}.$ Now my question is how many entire functions $g$ are there such that $f=g$ for $|z|>1$?According to me there is no such ...
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1answer
35 views

How to show that any path $\gamma:[a, b]\rightarrow\mathbb C$ is rectifiable and that $L(\gamma)=\int_{a}^{b}|\gamma'(t)|dt$.

How to show that any path $\gamma:[a, b]\rightarrow\mathbb C$ is rectifiable and that $L(\gamma)=\int_{a}^{b}|\gamma'(t)|dt$. Definition: Consider a partition $P,a=t_0\lt t_1\lt \ldots\lt t_n=b\; ...
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3answers
49 views

Integration of a real valued function on complex plane

Suppose $f: \mathbb{C}\rightarrow \mathbb{R}$ $f$ is continuous, bounded, $f(z)\geq 0$. Can we claim that the following integration $$\int_{C_R}f(z)dz$$ is equal to zero? ($C_R$ is a circle ...
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1answer
32 views

Jordan curve of infinite length

I was thinking about Jordan curve with infinite length and Koch snowflake seems to be a valid answer intutively. Can anyone give mathematical proof for this?
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1answer
14 views

Prove $h=$Log $z$ in a domain

I'm trying to show that if $h$ is analytic in $D=\{z\in\mathbb{C}: |z-1|<1\},$ $h'(z)=z^{-1},$ and $h(1)=0,$ then $h$ is Log $z$ in $D$. I know Log' $z=z^{-1}$ and Log $1=0.$ However, I don't know ...
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1answer
52 views

Complex Analysis Lectures

I am looking for a series of video lectures on the subject of complex analysis which follow Conway's text Functions of Complex Variable I. Any recommendations?
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13 views

Limit argument for complex squareroot

Let $z\in \mathbb{H}\backslash(0,i]$, where $\mathbb{H}$ is the upper half plane. I want to show that $z(\sqrt{z^2+1}-z)\rightarrow \frac{1}{2}$ for $z\rightarrow \infty$, where $w\mapsto \sqrt{z}$ ...
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12 views

Optimizing functions with a complex domain and a real codomain

In general I want to understand if it makes sense to optimize a function of the following form $f: \mathbb{C} → \mathbb{R}$ for my specific problem $f(z) = | z | ^{2} $ (wich I is not analytic since ...
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0answers
24 views

technique of proving continuity complex function

If I want to prove a function $f(t)$ is continuous at $ t_o$. By def of continuity, I need to show that for $\epsilon >0 $, there exist $ \delta>0$ s.t $ |t-t_o| < \delta$ implies $ ...
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0answers
31 views

Describe set of $z^2$ as z moves over 2nd quadrant and show it is open and connected

Problem: Describe the set of points $z^2$ as $z$ varies over the second quadrant: {z = x + iy; x < 0 and y > 0}. Show this is an open connected set. (Hint: use the polar representation of z.) The ...
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0answers
21 views

Zeros of this function?

Let $$f(z)=\gamma + z^{\beta_2-\beta_1}$$ where $\gamma\in \mathbb{R}$, $\beta_1\in \mathbb{Z}$, $\beta_2 \in \mathbb{Z}$ and $\beta_2 > \beta_1$. The variable $z$ takes complex values. Is there a ...
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4answers
56 views

Laurent series of $\dfrac{1}{\sin(\frac{1}{z})}$ [on hold]

What is the Laurent series for the function $f(z)=\dfrac{1}{\sin(\frac{1}{z})}$ at the point $z=\dfrac{1}{\pi}$ and $z=0$?
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0answers
31 views

Problem with $\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$ (by residues) [duplicate]

I, I am trying solve the following integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$$ Teachers teached me that I can solve the integral $$\int_{0}^{\infty} ...
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0answers
16 views

Find the radius of convergence [on hold]

Find the radius of convergence of $\sum\limits_{n=0}^{\infty} \exp(inθ)(θ^n)z^{2n}$ with n running from 0 to infinity, non-zero θ. I would usually apply the ratio test, but with θ in the equation too ...
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0answers
28 views

Fourier coefficients of the Gaussian.

I would need to find the fourier coefficient of this gaussian for a problem. I'm now stuck with this integral, \begin{equation} c_{n}=\int_{-1}^{1}e^{\frac{x^{2}}{2}}\left(\cos\left(\pi ...
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30 views

Homotopic, winding number and continuity, Conway text

I have a question about this. Here is the def of homotopy used by Conway, in case you guys need it. Let $G$ be an open set in $\mathbb{C}$ and let $\gamma$ be a closed smooth rectifiable curve in ...
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2answers
20 views

Show that $\lim_{z \to 0} \frac{\Re(z)}{z}$ doesn't exist.

Show that $\lim_{z \to 0} \frac{\Re(z)}{z}$ doesn't exist. Let $z=r(\cos(\theta)+i \sin(\theta))$. So $\frac{\Re(z)}{z} =\cos ^2(\theta) - i \cos(\theta)\sin(\theta) $, and $$\lim_{z \to 0} ...
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0answers
39 views

If there is a branch of $\sqrt{z}$ on an open set $U$ with $0 \notin U,$ then there is also a branch of $arg$ $z.$

Show that if there is a branch of $\sqrt{z}$ on an open set $U$ with $0 \notin U,$ then there is also a branch of $arg$ $z.$ I am unable to proceed any further in this and any help in this regard ...
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3answers
52 views

Can a complex argument into this function ever yield a real result?

I have a function defined as: $f(z)=\frac{\Gamma{(z)}+1}{z}$ Are there any $z ∈ C$ (with nonzero imaginary part) such that $f(z)∈R$? I tried substituting in $z=a+bi$ with $b≠0$ into the product ...
2
votes
4answers
48 views

Analyticity of $\overline {f(\bar z)}$ given $f(z)$ is analytic [duplicate]

Suppose $f$ is an analytic function on a domain $D$. Then I need to show that $\overline {f(\bar z)}$ is also analytic. Here is what I did - Suppose $f(z) = u(x,y) + iv(x,y)$ where $u$ and $v$ are ...
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1answer
40 views

Complex integration on upper-half plane

In order to prove the normalisation property of a Lorentzian function, $L = \dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$ we take a closed contour on the ...
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1answer
36 views

Determine for what values $z \in \mathbb{C}$, $\sum_{n = 1}^{\infty} \frac{z^n}{n^2}$ is convergent.

I am not sure where to start on this one. I know that $z^n$ can be written as $\sum_{n=0}^{\infty} \frac{1}{1-z}$. But I do not know how to proceed.
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0answers
19 views

Branch point at infinity?

I have to find the branch points of $f(z)=\left( z(z+1)\right )^{1/3}$. It is clear that $0$ and $-1$ are branch points, but I am not sure about infinity. Making the substituition $x=\frac{1}{z}$ and ...
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2answers
47 views

Does there exist an analytic function $f$ on $D(0,1)$ such that $f(z_n)=0$ for even $n$ and $f(z_n)=1$ for odd $n$?

Given that $(z_n)$ is a sequence of distinct points in $D(0,1)=\{z \in \Bbb C : |z| \lt 1\}$ with $\lim_{n \to \infty} z_n=0$, Can we find an analytic function $f$ such that $f(z_n)= \begin{cases} 0, ...
0
votes
2answers
45 views

Is $f( x + iy) = e^{-x} e^{-iy}$ complex - differentiable?

I started by letting $u(x,y) = e^{-x}$ and $v(x,y) = e^{-iy}$ . I then tried to use the cauchy reiman equations : $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial ...
0
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2answers
46 views

Let $\gamma=\{z\in \Bbb C: \lvert z \rvert=2\}$ in anti-clockwise orientation. Then $I=\frac {1}{2\pi i} \int_{\gamma} z^7 \cos \frac 1{z^2} dz$=?

$$I=\frac {1}{2\pi i} \int_{\gamma} z^7 \cos \frac 1{z^2} dz=?$$ The function $\cos \frac 1{z^2}$ is neither analytic at $z=0$ and nor it has a pole at $z=0$. By Cauchy Integral Formula can I get ...
0
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1answer
32 views

Show that $g(z)=\frac{1}{n}\sum_{k=0}^{n-1} f \left(\xi^{k}\sqrt[n]{z}\right)$ is an entire function.

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f(\xi z)=f(z)$ for all $z\in \mathbb{C}$ and consider the ...