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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47 views

Need help about harmonic functions!

I have trouble on solving the following problem: Show that there doesn't exist a non-constant function $u$ such that $u$ is harmonic on C and for $z=x+iy$ in C that $u(z)>4x^2+9y^2+1$.
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1answer
89 views

Integrating this complex function, using Residue Theorem [duplicate]

I am having a massive amount of trouble integrating this, I really have no clue how to get the answer in the book: $$\int_{-\infty}^{\infty} \frac{x^4}{1+x^8}dx$$ I know I need to find the poles ...
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1answer
156 views

Complex number with 3 dimensions [duplicate]

I was looking back on complex analysis and asked myself: ''Why is there no complex number in 3 dimensions ?''. To place this question let me define with what I mean with 3 dimensions in the following. ...
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1answer
59 views

Taylor Series & complex analysis

I am taking complex analysis. There's a question in the book when trying to prove the theorem, and the theorem goes like this: If $f$ is analytic in the disk $|z-z_0|<R$,then the taylor ...
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2answers
484 views

What is a reference for the ( classical and well-known ) proof of Weyl's lemma?

What is a reference for the (classical and well-known) proof of Weyl's lemma that states: Let $U$ be an open subset of $R^n$. Then if $f\in L^1_{loc} (U)$ and if $\int_U ...
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1answer
40 views

$\frac{\operatorname{Log}w}{w-1}$ is analytic

I'm trying to show that $g(w)=\frac{\operatorname{Log}w}{w-1}$ when $w \neq 1$ and $g(1)=1$ is analytic when $0<|w|<\infty$ and $- \pi < Arg (w) < \pi$. I started out by finding the ...
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2answers
736 views

condition for roots of quartic equation to be purely imaginary

(a) Show that the roots of equation $z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ where $a_1, a_2, a_3, a_4$ are real constants different from zero, has a pure imaginary root if $a_3^2 + a_1^2 a_4 = ...
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41 views

Where does infinity go in this theorem?

I'm currently studying Mobius transformation and i have proved the following: (I formulated this sentence by myself, so if this approach doesn't seem appropriate please suggest me a better ...
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1answer
266 views

One-one analytic functions on unit disc

Is the following statement true? Suppose, $ f:D\to \mathbb C $ is an analytic function where $ D $ is the unit disc of radius $ 1 $ around $0 $. Suppose, $ f $ is analytic on the boundary of $ D $ as ...
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1answer
33 views

How do I find zeros in D(0,2)

$p(z) = z^8 - 20z^4 + 7z^3 + 1$. I know there is 4 real roots, but how do i figure out how many zeroes are there in $D(0,2)$?
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358 views

Why is the polynomial $S(\vec{x})$ with coefficients obeying a constraint homogeneous?

I have recently been working on a problem to prove that a particular polynomial is in fact homogeneous. Although I have found out that this is true, I am curious to see whether there might be a deeper ...
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2answers
37 views

Cauchy Riemann Equations valid for $U$

Show that if $f: U \to \mathbb{C}$ is analytic on an open set $U$ in $\mathbb{C}$, then the Cauchy Riemann equations for $f$ hold in $U$. My question is, how do the Cauchy-Riemann equations differ ...
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1answer
49 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
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2answers
126 views

Determine the number of zeros in the first quadrant

This is a homework question: $$f(z) = z^2 - z + 1$$ sorry for the poor code!
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1answer
36 views

How to integrate in s-domain

We know that a simple integral in the time domain for example, is put like that: $$\int_a^bf(t)dt$$ So $t$ will vary from $a$ to $b$ and all values in between (like a straight line). But when I do ...
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1answer
121 views

a sequence of polynomials converges to $0$

I am trying to show that there is a sequence $(P_{n})_{n}$ of polynomials such that $P'_{n}(0)=1$ for all $n$, $P'_{n}(z)\rightarrow0$ if $z \in \mathbb{C}^{\times}$ and $P_{n}(z)\rightarrow0$ ...
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3answers
230 views

Determine the number of zeros in the first quadrant $f(z) = z^4- 3z^2 + 3$ [closed]

Determine the number of zeroes of the following function which are in the first quadrant: $$f(z) = z^4- 3z^2 + 3$$ Help please!!! I'm not that good at complex variables!
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2answers
97 views

What is the well-known formula?

I am reading a paper and I puzzled with the following formula : Suppose $g\in C^2 (R^2)$ with compact support,show $$-\frac{1}{2 \pi} \iint_{R^2} \Delta g(z) \log \frac{1}{|z-\xi |}dxdy =g(\xi)$$ I ...
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1answer
92 views

Question relating to the Casorati-Weierstrass Theorem.

The question I am trying to answer is: Suppose $f$ is analytic in the punctured disc $0 < |z| < 1$ except for poles $\{z_n\}$ where: $$\lim_{n \to \infty}z_n = 0$$ Note that $0$ is not an ...
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2answers
83 views

Laurent Series and Residue

Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point. $${z^2 \over (z^2-1)}; \ \ z=1$$ I've broken the function into two ...
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1answer
170 views

nth derivative of a holomorphic function

I am to show that if f(z) is holomorphic within (and on) a circle C with radius r and center a, and if |f(z)|<= M for all z in C, that the nth derivative of f at $a$ is less than or equal to ...
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4answers
48 views

How do i prove this set has at most 2 elements?

Let $w,\alpha\in\mathbb{C}$ and $\delta,\epsilon >0$ such that $(w,\delta)\neq (\alpha,\epsilon)$ Define $G=\{z\in\mathbb{C} : |z-\alpha|=\epsilon \text{ and } |z-w|=\delta\}$ How do i prove that ...
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1answer
82 views

Bounded entire function constant

if $f$ is entire and $f(z) < |R|\varepsilon(R)$ for large $|z| = R$ where $\varepsilon\rightarrow0 $ as $R\rightarrow\infty$, show $f(z)$ is constant. I have used $|g(z)| = |f(z) - f(0)|/z$ and ...
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83 views

Very quick question: function extends to 1/z on the boundary of unit disc

How can one show there is no holomorphic function $f$ on the open unit disc $\mathbb{D}$ such that it extends continuously to $\frac{1}{z}$ on $\partial\mathbb{D}$? I mean $f$ takes value ...
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1answer
71 views

Complex integration, help

I need help integrating $\int_{-\infty}^{\infty}\frac{z \sin (z)}{\left(z^2+1\right) \left(z^2+2\right)} dz$. I calculated the integral over the closed upper half circle in the complex plane which is ...
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1answer
37 views

Why is this a line equation?

Define $$L=\{z\in\mathbb{C} : cz + \overline{cz} + w = 0\}$$ Where $c$ is a nonzero constant. How does $L$ represent a line?
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1answer
72 views

Contour integral of $\frac{\bar{z}}{z-Z}$ on a square centered at the origin

I am having trouble calculating the following integral: $\oint_C \frac{\bar{z}}{z-Z} dz$ Here, Z is a complex constant and C is the contour of a square of side $2a$ centered at the origin. I ...
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2answers
90 views

Find $\int_\gamma \frac{dz}{z^2}$ wihtout explicit calculations

Evaluate the following integral without doing any explicit calculations: $\int_\gamma \frac{dz}{z^2}$ where $\gamma(t) = \cos(t) + 2i\sin(t)$ for $0 \le t \le 2\pi$. This exercise comes along with ...
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365 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
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2answers
51 views

Integral along $\Gamma_c := \{c + i t \mid c>0 , -\infty < t < \infty\}$

I have a Complex Analysis homework problem which I've been working on for some time, and have become stuck. I am asked to compute $$ I \equiv {1 \over 2\pi{\rm i}}\int _{\Gamma_c}{a^{s} \over ...
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2answers
803 views

Non-existence of a bijective analytic function between annulus and punctured disk

Suppose $A=\{z\in \mathbb{C}: 0<|z|<1\}$ and $B=\{z\in \mathbb{C}: 2<|z|<3\}$. Show that there is no one -to-one analytic function from A to B. Any hints? Thanks!
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1answer
190 views

Prove this integral is zero

I'm trying to prove that $\lim_{R\to\infty}\int_{C_R} dz \exp\left(iaz^2\right) = 0$, where $a$ has a positive imaginary part and $C_R$ is an arc from $R$ to $\frac{1+i}{\sqrt{2}}R$ along the circle ...
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1answer
189 views

a sequence of holomorphic functions with uniformly convergent derivatives

Let $(f_{n})_{n}$ be a sequence of holomorphic functions on a domain D which satisfies the following conditions: there exists some $z_{0}$ in D such that $f_{n}(z_{0})$ converges and the sequence of ...
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1answer
102 views

Showing $\operatorname{Log}(z-i)$ is not analytic

Show that the function $\operatorname{Log}(z-i)$ is analytic everywhere except on the half line $y=1$ $(x\leq 0)$. I know that ...
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2answers
51 views

What is the definition of 'line' in $\hat{\mathbb{C}}$?

What is the definition of straight line in $\hat{\mathbb{C}}$? Is it defined as $\{x\in\mathbb{C}: \frac{Re(x-a)}{Re(b)} = \frac{Im(x-a)}{Im(b)}\}\cup \{\infty\}$? ($a,b$ are complex numbers and ...
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1answer
108 views

Whats the differences between the real-entire functions on $\mathbb R^{2}$ and complex entire functions on $\mathbb C$?

We note, as set of points, $\mathbb R^{2}= \mathbb C.$ A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point ...
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0answers
57 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
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1answer
158 views

Is it possible for a function to be differentiable at only one point?

I was taking a complex analysis class today, and we looked at the function f(z)=|z|^2 (with the domain over the complex numbers). It is continuous, but it satisfies the Cauchy-Riemann equations at ...
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1answer
39 views

Complex Analysis Computation

I'm not really sure how to tackle this problem, so any help/hints would be appreciated. Let $w=\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ where n is a positive integer. ...
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1answer
36 views

Conformal mapping on two paths

GIven $ f (z) = z^2$ . Let $p = (0, −1)$ and take the curves $γ_1, γ_2$ passing through $p$ as $γ_1 = $arc of the unit circle through $(0, −1) $ counterclockwise and $γ_2 $ = a straight line ...
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Find a branch of $f(z)= \log(z^3-2)$ that is analytic at $z=0$.

Find a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$. Can anyone help me on this question? I have no idea how to find a branch. The definition of branch given in lecture is $F$ is a branch ...
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2answers
84 views

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic?

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic? If yes, in which domain?
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0answers
25 views

Value of a transcendental function

I am trying to transform really just evaluate or simplify: Log($\sinh(1+i))$ into Log($\sinh^2(1) + \sin^2(1)) + i\theta$ where $\theta=\arctan(\frac{\tan(1)}{\tanh(1)})$ I have tried the ...
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1answer
63 views

Is there a name for the one-point compactification of $\mathbb{C}$?

Let $\hat{\mathbb{C}}$ be the one-point compactification of $\mathbb{C}$. This space $\hat{\mathbb{C}}$ is called the Riemann sphere. If I want to designate the topology $\tau$ on ...
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2answers
59 views

Show $e^z e^w = e^{z+w} \ \forall \ z,w \in \mathbb C$ by differentiation of $f(t):=e^{w+tz}e^{-tz}, \ t \in \mathbb R$.

Show $e^z e^w = e^{z+w} \ \forall \ z,w \in \mathbb C$ by differentiation of $f(t):=e^{w+tz}e^{-tz}, \ t \in \mathbb R$. I have already showed $e^z e^{-z} = 1$ for $z \in \mathbb C$. This result ...
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1answer
27 views

Finding residue with respect to closed path

Let $\gamma$ be the closed path consisting of straight line segments from $2+2i$ to $-2-2i$, from there to $-2+2i$, from there to $2-2i$ and finally back to $2+2i$. Evaluate $\int_{\gamma}f(z)dz$ for ...
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1answer
58 views

Showing $f$ must be constant on $\mathbb{C}$ given 3 conditions

Suppose we know the following about a function $f(z)$. i. $f(z+1)=f(z)$ and $f(z+i)=f(z)$ for all $z$ in $\mathbb{C}$. ii. $f$ has only isolated singularities (if any) in $\mathbb{C}$ iii. $f$ has ...
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1answer
225 views

Residue theorem in evaluating complex integrals?

It's been a while since I used residue theorem to evaluate anything. I remember that whenever we have a real valued function, we can use residue theorem to evaluate its integral with associated ...
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1answer
48 views

hyperbolic inequality

Calculating some contour integral, I have to prove that $\int^{R+i}_{R}\frac{cosh(az)}{cosh(\pi z)}dz$ goes to zero if R goes to infinity. And we know that $\left|a\right|<\pi$. I want to use the ...
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2answers
39 views

Proving $-u$ is a harmonic conjugate for $v$

Suppose $u$ and $v$ are real valued functions on $\mathbb{C}$. Show that if $v$ is a harmonic conjugate for $u$, then -$u$ is a harmonic conjugate for $v$. I know I have to use cauchy reumann here. ...