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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1answer
12 views

Dettman's Applied Complex Variables Theorem 4.2.5 Correction (Normal Families)

I need someone to decide whether I'm going crazy. Dettman states the following theorem without proof: Theorem 4.2.5 Let $F$ be a family of functions analytic in a domain $D$, where it is uniformly ...
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1answer
32 views

Is order of poles of functions determined by Laurent series?

Suppose $$f(z) = \frac{1}{(z-2)^5z}$$ is given. By looking function, i will tell there is a $5$th-order pole at $z=2$ which is in fact true. But on the other hand suppose $$f(z) = \frac{\sin ...
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21 views

Show that if $u_n (z_0) \rightarrow 0$ for some $z_0 \in D$, then $u_n \rightarrow 0$ uniformly on compact subsets of $D$.

Let $D \subseteq \Bbb C$ be a connected open subset and let {$u_n$} be a sequence of harmonic functions $u_n: D \rightarrow (0,\infty)$. Show that if $u_n (z_0) \rightarrow 0$ for some $z_0 \in D$, ...
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0answers
49 views

Why the proof isn't complete?

I'm going through some complex analysis exercises and found one with which I have some problems: For all real $y$, $$\int\limits_{-\infty}^\infty e^{-(x+iy)^2}dx = \int\limits_{-\infty}^\infty ...
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21 views

A convergent infinite product

Prove that $\Pi_{n=1}^{\infty}\,\,\frac{|a_{n}\,|}{a_n} \,(\frac{a_{n}-z}{1- \overline{a_{n}}\,\,z\,}\,)$ converges in $H(B(0,1))$ if and only if $\sum_{n=1}^{\infty}\, \,(1-|a_{n}|)< \infty$. I ...
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0answers
11 views

A sequence of polynomials covering to an analytic function on a given region.

Let $G=\{z: |z|<1 \text{and} \,\, |z-\frac{1}{2}|> \frac{1}{2}$}. If $f$ is analytic in $G$, explain why there exists a sequence $\{p_{n}\}$ of polynomials which converge uniformly to $f$ on ...
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4answers
535 views

How can I simplify this complex number to get a real number?

$$\large \frac {e^{i \frac{\pi a}{2}}[1-e^{i\pi a}]} {[1-e^{i2\pi a}]}$$ I am trying to arrive at $$\frac {1}{2\cos\left(\frac{\pi a}{2}\right)}$$ I've tried dividing top and bottom by one of the ...
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2answers
35 views

Showing an analytic function on the unit disk is identically zero

Suppose that $f$ is analytic on the open unit disk and there is a constant $M > 1$ such that $|f(1/k)| \leq M^{-k}$ for $k \geq 1$. Show that $f$ is identically zero. I see that $f(0) = 0$, that ...
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0answers
34 views

Understanding “asymptotic” behaviour of a set

In a book, an author defines a complex polynomial \begin{align*} p(z)=\alpha_{n}z^{n}+\alpha_{n-1}z^{n-1}+\cdots+\alpha_{1}z+\alpha_{0},\quad \alpha_{n}\neq0, \end{align*} $n\in\mathbb{N}$ such that ...
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1answer
43 views

An analytic function $f$ bounded on the right half plane and $|f(z)|\leq 1$ on the imaginary axis

Assume that $f$ is an analytic function that $|f(z)|\leq 1$ on the imaginary axis and that $f$ is bounded in the right half plane. Prove that in fact $|f(z)|\leq 1$ in the right half plane. Hint: ...
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1answer
22 views

Zero of holomorphic function

Let $\Omega \subset \mathbb{C}$ be an open set that contains the unit ball $D$ and let $f \in \mathcal{O}(\Omega)$ a non constant map s.t. $|f(z)| = 1$ for all $z \in \partial D$. Show that $f$ has a ...
2
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2answers
46 views

What is the interval for possible values of the argument of a complex number?

It looks like there are different intervals in which the argument of a complex number can be. Some say it goes from $-\pi$ to $+\pi$ others say it goes from $0$ to $2\pi$. For the most part, both ...
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1answer
39 views

Existence of holomorphic function from unit disc to itself.

Does there exists a holomorphic function from open unit disc to itself s.t. $f(1/2)=-1/2$ and $f'(3/4)=1$? I think the answer is 'Yes' as $f(z)=z-1$ satisfies these conditions. Kindly correct me if I ...
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0answers
17 views

If $f:B_N\rightarrow \mathbb{D}$ and $z_n\in B_N$ with $\{f(z_n)\}$ thin, is $\{f(\phi(z_n))\}$ thin for any autmorphism $\phi$ of $B_N$?

Let $B_N$ denote the open unit ball in $\mathbb{C}_N$. A sequence $\{z_n\}$ of distinct points in $\mathbb{D}$ is called thin if $\lim_{k\rightarrow \infty}\displaystyle\prod_{j: j\not =k}^\infty ...
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1answer
39 views

Estimates for $1/\zeta(s)$

Recently I am reading Stein's Complex Analysis, and he is going to prove the prime number theorem after estimating the value $1/\zeta(s)$. However, I don't understand the technical details of the ...
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16 views

show that exists exactly one n such that $\gamma_1$ is homotopic to $\gamma_2 = e^{2\pi int}$

Let $\gamma_1 $ be a closed path in $\Bbb{C}\setminus0$ such that $0 \in int (\gamma_1)$ show that exists exactly one $n \in \Bbb{Z}\setminus 0$ such that $\gamma_1$ is homotopic to $\gamma_2 : ...
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0answers
35 views

How to solve this equation with implicit sum

I want to know how the authors of this arxiv paper (p. 10) solved the equation \begin{align} g\left(\lambda\right) ={}& ...
2
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0answers
38 views

Expansion of Weierstrass elliptic function in second period

I would very much like to find expansions of the Weierstrass $\wp$ and $\zeta$-functions for small absolute values of the second period $\omega_2$. So, more precisely, I would like, for ...
2
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1answer
82 views

My complex integral cancels at the end; how can I modify the integrand to prevent this?

$$\int_0^\infty \frac{x^a}{x^2 + b^2}$$ for $-1< a < 1$ and b>0 -- these constraints help with estimating the integral on the big circle and small circle of a keyhole contour that I chose to ...
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1answer
32 views

Investigate the properties of complex function $f(z)=\frac{z}{\sqrt{1+|z|^2}}$ where $f:\mathbb{C}\rightarrow \mathbb{D}$

Investigate the properties of the complex function $f(z)=\frac{z}{\sqrt{1+|z|^2}}$ where $f:\mathbb{C}\rightarrow \mathbb{D}$ I am required to prove that it is bijective, continuous and find the ...
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2answers
53 views

Show that $\int_{|z|=1}\,\,\,(1+z)^{\alpha} dz=0$

Consider the principal branch of $f(z)=(1+z)^{\alpha}$, where $\alpha$ is real. Show that for $\alpha > -1$ $\int_{|z|=1}\,\,\,(1+z)^{\alpha} dz=0$. Now the integrand doesn't have an ...
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0answers
57 views

Evaluate $\frac{1}{\pi i\,\,} \int_{\alpha} \frac{\cos z}{2\sin z-\sqrt{2}\,\,\,\,\,\,}dz$

Let $D=\{z:\frac{\pi}{4}<|z|< \frac{5\pi}{4}\,\}$ and let $\alpha$ be a closed curve in $D$ whose winding number about the origin is $2$. Evaluate $$\frac{1}{\pi i\,\,} \int_{\alpha} ...
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3answers
129 views

Show that $\displaystyle{\int_{0}^{\infty}\!\frac{x^{a}}{x(x+1)}~\mathrm{d}x=\frac{\pi}{\sin(\pi a)}}$

Show that for $0<a<1$ $$\int_{0}^{\infty}\frac{x^{a}}{x(x+1)}~\mathrm{d}x=\frac{\pi}{\sin(\pi a)}$$ I want to solve this question by using complex analysis tools but I even don't know how to ...
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4answers
139 views

Can I use an upper semi-circle to integrate this function?

I'm trying to integrate $$\int_{-\infty}^{\infty} \frac{e^{iz}}{e^z + e^{-z}}dz$$ Do I have have to integrate this over a box, or can I use my first guess at a contour and use an upper semi-circle ...
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0answers
15 views

Q: Dirichlet Problem

I have to show that $\psi(x,y) = \pi + 2Arg(z)$, $\ -\pi < Arg(z) < \pi$, is a solution of the DP on the upper half plane $[{(x,y) \in \Re^2: y > 0}]$ with limiting value $\pi$ on that part ...
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1answer
37 views

Showing a function analytic on open unit disk can be analytically extended

Suppose that $f$ is analytic on the open unit disk such that there exists a constant $M$ with $|f^k(0)| \leq k^4M^k$ for all $k \geq 0$. Show that $f$ can be extended to be analytic on $\mathbb{C}$. ...
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0answers
64 views

Contour integral mystery: why is the answer different from Maple/Matlab?

The mystery is that here is a fairly standard contour integral which can be done by the residue theorem. Yet when I tried to evaluate it using numerical softwares like Maple or Matlab, the answer is ...
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2answers
138 views

Evaluate $\int_0^\infty \frac{\sqrt{x}}{x^2+1}\log\left(\frac{x+1}{2\sqrt{x}}\right)\;dx$

Prove that $$\int_0^\infty \frac{\sqrt{x}}{x^2+1}\log\left(\frac{x+1}{2\sqrt{x}}\right)\;dx=\frac{\pi\sqrt{2}}{2}\log\left(1+\frac{\sqrt{2}}{2}\right).$$ I managed to prove this result with some ...
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0answers
8 views

What is the branch point of $f(z)=\tanh^{-1}(\frac{z}{\sqrt{(1+a^2)(z^2+1)}})$

What is the branch point of $f(z)=\tanh^{-1}(\frac{z}{\sqrt{(1+a^2)(z^2+1)}})$, where $a$ is small real number $a\ll1$. I know the branch point of ...
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0answers
40 views

How can one conclude that $f$ is differentiable?

How can one conclude that $f$ is differentiable ? If $f:\mathbb R\to S^1=\{z\in\mathbb C:|z|=1\} $ is continuous and such that $f(a+b)=f(a)f(b)$ then the formula holds $\displaystyle ...
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0answers
24 views

Find the multiplicity of $z_0$ as a zero of the composition function $f\circ g$

Suppose $f$ and $g$ are analytic in a neighborhood of $z_0$ , $f(z_0)=0$ with multiplicity $m$ , $g(z_0)=0$ with multiplicity $n$. What is the multiplicity of $z_0$ as a zero of the composition ...
4
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6answers
213 views

$\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$ [duplicate]

How can I show that $\arctan (x) + \arctan(1/x) =\frac{\pi}{2}$? I tried to let $x = \tan(u)$. Then $$ \arctan(\tan(u)) + \arctan(\tan(\frac{\pi}{2} - x)) = \frac{\pi}{2}$$ but it does not ...
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4answers
64 views

Factorization of a polynomials in complex number.

Factorize this expression: $$a^2+b^2+c^2-ab-bc-ca.$$ The result is $$(a+b\Omega+c\Omega^2)(a+b\Omega^2+c\Omega)$$ How I can get $\Omega$ here?What's the approach?
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3answers
60 views

Is this a mistake in the book: de Moivre applied to real number is multip valued?

Consider $$ z^n = \rho^n (\cos (n \theta) + i \sin (n \theta))$$ for $n \in \mathbb Z$. It is written in the book I am reading that this is multi-valued when $n$ is replaced by a real number. ...
2
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0answers
53 views

Prove that this infinite product converges uniformly,

Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers such that (i) $0<|a_n|<1$, (ii) $\sum_{n=1}^\infty(1-|a_n|)<\infty$ Prove that the infinite product $$\prod_1^\infty \frac{(a_n ...
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0answers
18 views

Deforming path of integration from the real line to the boundary of a open subset of the upper half complex plane.

Denoted the upper half of the complex plane by $\mathbb{C}^{+}=\{z\in\mathbb{C}:\text{Im }z>0\}$. Let the open, unbounded set $A\subseteq\mathbb{C}^{+}$ have a boundary $\partial A$ such that the ...
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53 views

Show that the series $\sum_{n=1}^{\infty}\,\, \frac{z^n}{1+z^{2n}}$ converges

Show that the series $\sum_{n=1}^{\infty}\,\, \frac{z^n}{1+z^{2n}}\,\,\,$ converges in both interior and exterior of the the unit circle and represents an analytic function in each region. I want to ...
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1answer
37 views

Bijective Conformal Mapping onto the Open Unit Disc $\mathbb{D}$

What is the explicit bijective conformal mapping $f(z):G_n\to\mathbb{D}$, $z\in\mathbb{C}$ for the following domain transformations: $G_1=\{x+iy~|~x>1/2,y>0\}$ is the open region of the first ...
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4answers
334 views

The Riemann Sphere Interpretation

Is the Riemann sphere anything more than a simple visual tool to help students understand the complex planes, or the behavior of complex valued functions at infinity, limit points etc? Or is there a ...
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1answer
19 views

Showing an analytic function is exactly 0 or never 0 on a domain

Suppose that $f_n$ is a sequence of analytic function on a connected, open set $U \subset \mathbb{C}$ such that $f_n \to f$ on compact subsets of $U$. If $f_n(z) \neq 0$ for all $n$ with $z \in U$, ...
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3answers
41 views

Find a harmonic function in the first quadrant,

Find a harmonic function $\phi$(x,y) in the first quadrant with the boundary values $\phi$(x,0) = -1 for x>0, and $\phi$(0,y) = 1 for y>0. Is this function unique? My attempt was this: Consider ...
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1answer
21 views

Constant rank for an analytic matrix

Given a singular $n\times n$ matrix whose coefficients are analytic functions of a single variable in a neighbourhood of 0, I need to prove that there is an integer $r$, $1\leq r\leq n$ such that ...
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0answers
29 views

Prove that there exists an analytic function $g$ such that derivative takes maximum

In the complex comp this afternoon, I got stuck in this problem. Let $\mathcal{F}$ be a family of function mapping $\{z\mid\Re z>0\}$ to itself and $f(1)=1$. Prove that there exists a function ...
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0answers
40 views

If $f$ is one to one show that $f(a) \in \partial \Omega$

Let $G$ be a region. Let $a \in G$. Suppose that $f:(G-{a}) \to \mathbb{C}$ is an analytic function such that $f(G-{a})=\Omega$ is bounded. i) Show that $f$ has a removable singularity at $z=a$ ii) ...
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0answers
31 views

Analytic functions on $\mathbb{H}$ such that $f(i)=3i$

Let $\mathcal{F}$ be the set of analytic functions from the upper half plane to itself satisfying $f(i)=3i$. Find $\displaystyle \sup_{f\in \mathcal{F}} |f'(i)|=s$. Then either construct a function ...
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1answer
23 views

Why does the valence formula imply $M_k=0$ for $k <0$?

I'm studying Modular Forms and in the notes I'm reading the author states the following result, known as the valence formula: "Let $f$ be a non-zero weakly modular meromorphic form of weight $k$ ...
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1answer
15 views

Partition of unity subordinate to cover of subset of $\mathbb C$ in Hormander

In Hormander's Introduction to Complex Analysis in Several Variables, I'm confused by the usage of subordinate partitions of unity in the proof of a strengthening of the Mittag-Leffler theorem. In ...
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0answers
29 views

Solutions to the equation $\langle z,u_1\rangle \langle z,u_2\rangle =\langle u_1,u_2\rangle$.

Suppose $u_1$ and $u_2$ are elements of $\mathbb{C}^N$ of norm $1$, and that $\langle u_1,u_2\rangle\not =0$. If $z$ is in $\mathbb{B}_N$ (the unit ball of $\mathbb{C}^N$), how many solutions does ...
0
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1answer
25 views

$\Delta:$ $H^{\infty}(\Delta)$- domain of holomorphy

How to prove that $\Delta$ is an $H^{\infty}(\Delta)$- domain of holomorphy (i.e. there is a bounded holomorphic function $f\in H(\Delta)$ such that $\partial \Delta$ is the natural boundary). ...
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1answer
48 views

Solve complex equation $5|z|^3+2+3 (\bar z) ^6=0$

I'm stuck in trying to solve this complex equation $$ 5|z|^3+2+3 (\bar z)^6=0$$ where $\bar z$ is the complex conjugate. Here's my reasoning: using $z= \rho e^{i \theta}$ I would write $$ ...