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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Using ellipse for Cauchy integral.

I am currently working on a homework problem where I need to evaluate the following: $\int_{\gamma}(z^2+2iz)^{-1} dz$ where $\gamma(t) = 2\cos t+i \sin t, 0\le t\le 2\pi$ I was thinking of using the ...
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25 views

Solve $e^{\sin(z)}=1$ in the complex plane

I am trying to solve $e^{\sin(z)}=1$ in the complex plane. I know that this means that $\sin(z)=2k\pi i$ for some integer $k$. This is equivalent to saying that $$\frac{e^{iz}- e^{-iz}}{2i}=2k \pi ...
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17 views

Prove that any holomorphic map between Riemann surfaces is either constant or an open map

Part a) Prove that any holomorphic map $\textit{f: X $\rightarrow$ Y}$ between Riemann surfaces, with $\textit{X}$ connected, is either constant or an open map, meaning f(any open set) is open. Hint: ...
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1answer
29 views

How to conclude this proof that real and imaginary parts of holomorphic functions are harmonic.

I want to prove that if $f$ is holomorphic on an open set $\Omega$, then both the real and imaginary parts are harmonic, so I have proved that: $$4\frac{\partial}{\partial z} \frac{\partial}{\partial ...
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12 views

Curl of a vector in the complex plane

Let there be a vector $u(z)$ in the complex plane. Are these two statements equivalent? $$\nabla\times\overline u=\overline{\nabla\times u}$$ If not, why? I think they should be equal since $\nabla$ ...
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25 views

$R(\alpha) = \frac{f(\alpha)}{g(\alpha)}$ achieves $\beta \in \mathbb{C}, n$ times.

Let $R(\alpha) = \frac{f(\alpha)}{g(\alpha)}, \ \alpha \in \mathbb{C}$. $R$ is rational and $f(\alpha), g(\alpha)$ are polynomials with no factors in common. Define $n=$ deg$(R(\alpha)) = $ max deg ...
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1answer
36 views

Proof: Raising a complex number to a rational power

The problem from the textbook is: Prove that if (a complex number) $z$ is a number on the unit circle, then $z$ has finitely many distinct powers $z^n$ if and only if the argument of $z$ is a ...
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2answers
34 views

Proofs of Liouville Theorem

Are there proofs of Liouville theorem (bounded functions holomorphic in $\mathbb{C}$ are constants) without using the Cauchy theorem?
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2answers
50 views

The complex version of the chain rule

I want to prove the following equality: \begin{eqnarray} \frac{\partial}{\partial z} (g \circ f) = (\frac{\partial g}{\partial z} \frac{\partial f}{\partial z}) + (\frac{\partial g}{\partial \bar{z}} ...
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44 views

Is it true that $|e^z|\le e^{|z|}$ for all $z \in \mathbb C$?

Is it true that $|e^z|\le e^{|z|}$ for all $z \in \mathbb C$? I believe so as $x \le \sqrt{x^2+y^2}$ but the way the question is worded suggests it is not the entire complex plane.
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11 views

Mapping a simple region using an exponential function

I have the following region in the complex plane bounded by the two lines: $$ x = y \quad\text{and}\quad x = 2y$$ It is plotted as follows: Region Plot. I am required to map the region under the ...
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34 views

Mapping circles in the complex plane

I have the following two circles in the complex plane, $z=x+iy$, which bound a region, $R$. The equations for the circles are given as follows: $$x^2+(y−1)^2=1 \\ x^2+(y−2)^2=4 $$ That is, I believe, ...
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1answer
32 views

Locally bounded $\Leftrightarrow$ compactness

I am trying to prove the following A set $\mathcal{F}$ in $H(G)$ is locally bounded if and only if for each compact set $K\subset G$ there is a constant $$|f(z)|\leq M$$ for all $f$ in ...
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25 views

Complex integration with infinitely many poles on imaginary axis

I'm trying to integrate with a closed contour on the upper-half of the complex plane. $I = \displaystyle\int_{-\infty}^\infty \dfrac{z\,\mathrm{sech(z)}}{[(z-a)^2+b^2][(z+a)^2+b^2]} dz$ There are ...
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14 views

theorems or statement that guarentees a function to be a polynomial or constant specially complex functions

This is just a question out of curiosity since i came across that in complex analysis there is a lot of emphasis and stress to prove that certain function is const or a polynomial .can i learn some ...
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16 views

Conway, continuity and homotopy

Let $G$ be an open set in $\mathbb{C}$ and let $\gamma$ be a closed smooth rectifiable curve in $G$ such that $\gamma$ is homotopic to a constant curve,$\gamma_o$ ; furthermore let $ \Gamma(s,t): ...
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22 views

Function $\phi (x,y) \mapsto \phi (u,v)$ [ie:…] under the transformation $f(z) = u(x,y) + v(x,y)$ where $f(z)$ is analytic & $d_z \,f(z)$ is not $0$

I am just having a bit of trouble understanding what I am being asked. The ie: in title says $\phi [x(u,v), y(u,v)]$ Part a) is asking to do the laplacian, $\nabla^2_{x,y} \phi (x,y)$ and to write ...
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1answer
30 views

Prove that $ϕ ◦ ϕ = ϕ^2 = Id_{\Bbb C}$ (the identity map on C) if and only if $e^{iθ} \bar c + c = 0$.

Consider the isometry $ϕ : \Bbb C → \Bbb C$ with equation $ϕ(z) = e^ {iθ} \bar z + c$ where $θ ∈ \Bbb R$ and $c ∈ \Bbb C$. Prove that $ϕ ◦ ϕ = ϕ^2 = Id_{\Bbb C}$ (the identity map on C) if and only if ...
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1answer
31 views

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$.

Suppose that $f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$. I already have a proof for this but I would like an explanation ...
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2answers
31 views

Analytic function $f,$ such that $f(0) = 1$ and $f'(z) = zf(z),$ for all $z \in \mathbb{C}$

I'm trying to find an example of an analytic function $f$ satisfying the IVP $$ f'(z) = z\,f(z), \quad f(0) = 1, $$ and for all $z \in \mathbb{C}$, but I'm somewhat at a loss of the best way to ...
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19 views

Decomposing complex function in even and odd parts

It is known that for real functions, we can express any function $f(x)$ as a sum of an even function and an odd one, so that $$f(x)=f_e(x)+f_o(x)$$ where $$f_e(x)=\frac12[f(x)+f(-x)]$$ ...
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12 views

Growth of a sub-harmonic function is at least log.

Let $u$ be a subharmonic function, and let $A(u, r)$ denote the average value of $u(z)$ on $|z|=r$. I want to know if $A(u, r)>a\log(r)$ for large $r$. More generally, I want to know if $\lim_{r\to ...
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1answer
31 views

Evaluating the limit of a complex function

$$f(z) = \begin{cases} \displaystyle{\frac{\bar z^2+4}{z^2+4}|z+2i|^2},&z\neq\pm 2i\\ 0,&z = \pm 2i\end{cases}$$ 1. Evaluate $\displaystyle\lim_{z\to 2i} f(z)$ if it exists. 2. Evaluate ...
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21 views

Can a function be analytic if it is only differentiable on a line?

I was given an function and used the $CR$ equations to determine that $f$ was differentiable only when $y=2x$. Does this mean that it is nowhere analytic since there wouldn't be an ...
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3answers
86 views

If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a polynomial

I'm learning about complex analysis and need some help with this problem: If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a ...
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1answer
29 views

Proving a complex number is differentiable using the limit definition

Question: $f(x+iy) = xy^3$ Prove that $f(x+iy)$ is differentiable at $(0,0)$ My Attempt: $$\lim_{h\to 0}\frac{f(0+h,0+h)-f(0,0)}{h}.$$ $$\lim_{h\to 0}\frac{h^4-0}{h}.$$ $$\lim\limits_{h \to 0}{\ ...
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21 views

property of complex polynomials

I can't solve the following problem: Let $p(z) = z^n + a_{n-1}z^{n-1} + ... + a_0$ be a complex polynomial of degree $n \ge 1$. Assume that there exist $j \in \{0, 1, ... n-1\}$ such that $a_j \neq ...
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1answer
24 views

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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37 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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18 views

Help with understanding when Log(z^k)=k Log(z) as well as drawing the function.

For the question I'm dealing with the property Log(z^k)=k*Log(z)in which I have to find the largest open set that this property is true when $k$ is a positive integer. I understand that this ...
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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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1answer
37 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
32 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
16 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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3answers
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
15 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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22 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
15 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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21 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
89 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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35 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
47 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
14 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 $ ...
1
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1answer
27 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
42 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
30 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
51 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 ...