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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60 views

Existence of a Rectifiable Piecewise Smooth Path

Suppose you have $\gamma(t):[0,1]\rightarrow \mathbb{C}$ simple piecewise smooth, $\gamma(0) = 0$ and $\gamma(1)=1$. Does there exist $\eta:[0,1]\rightarrow \mathbb{C}$, another simple piecewise ...
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1answer
33 views

Showing a sequence of holomorphic functions converges uniformly on compact subset to a holomorphic function

Let $f_n$ be a sequence of holomorphic functions on an open, connected set $D \in \mathbb{C}$ with $|f_n(z)| \leq 1$ for all $z \in D$ and all $n \geq 1$. Let $A \in D$ be the set of all $z \in D$ for ...
3
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1answer
44 views

Question about Branch Cuts

I'm starting to learn a little complex analysis, and I'm a little confused as to what the purpose of a branch cut is. Is it to make a function continuous, or single valued? For example, the $\sqrt{}$ ...
3
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0answers
33 views

How can I calaculate this complex series? [closed]

Let $A=\{a+bi|a,b\in \mathbb{Z}\}-\{0\}$ For what $n$ does $\sum_{z \in A} 1/z^n$ $(n\in \mathbb{N})$ converge? If it converges, how can I evaluate it? Thanks in advance.
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1answer
25 views

Vector Surface integral of a Cube

Evaluate $${\int \int}_S \underline{F} \cdot d \underline{a}$$ $$\underline{F} = (y+z, x+z, y+x)$$ $S$ is the portion of the surface of the cube bounded by the planes $x=0, y=0, z=0, x=1,y=1,z=1$ ...
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2answers
34 views

A polynomial with two complex variables

Let $p(z,w)=a(z) + b(z)w +...+ t(z)w^k$, where $k\ge1$ and $a(z),b(z),...,t(z)$ are non-constant polynomials in the complex variable $z$. Then $\{(z,w)\in \Bbb C\times\Bbb C: p(z,w)=0\}$ is bounded ...
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2answers
46 views

Laplacian of $|f|^p,$ where $f$ is holomorphic

I have to prove that if $f$ is a homolorphic function that doesn´t vanish on its domain then $\triangle |f|^p=p^2 |f|^{p-2} |\frac{\partial f}{\partial z}|^2$ . My attempt: I take $|f|^p=(z ...
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0answers
28 views

Finding number of zeros

Find the number of distinct zeroes of $f(z) = z^6 + (10 - i)z^4 + 1$ inside $(-1,1)$ x $(-1,1)$. I know by applying Rouché's Theorem that $f$ has 4 zeros inside the given domain, but I'm not sure ...
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0answers
28 views

extensions of holomorphic function on the upper half plane

Let $f$ be a holomorphic function on the open upper half plane? Can $f$ be extended to the closed upper half plane, in general? Which holomorphic function can be extended to be holomorphic on the ...
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1answer
32 views

An isomorphism between two Banach algebras

Consider the compact set $[-1,1]$ and $C([-1,1])$ the set of all continuous functions $\phi: [-1,1] \rightarrow \mathbb{C}$. I want to show that the quotient of $C([-1,1])$ by $\mathbb{C}$ is the ...
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15 views

An exercise on the mulitplication of dirichlet series

This is an exercise problem in the chapter 7 of the Stein complex analysis book. I am stuck at (a). In fact, I have no idea how to deal with the condition "mk = n". Could anyone please help me?
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1answer
72 views

Generalization to this integral

$$ \int_0^\infty \frac{\ln(1 + x^a)x^s}{1+x^2} \ dx $$ Actually the problem was $ \displaystyle \int_0^\infty \frac{\ln(1 + x^a)}{(1+x^2)\ln(x)} \ dx $. But I guess the form of a Mellin Transform ...
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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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0answers
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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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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0answers
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
538 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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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 ...
2
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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 ...
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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 ...
2
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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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0answers
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 : ...
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
votes
1answer
83 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 ...
2
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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
58 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
131 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 ...
3
votes
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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votes
0answers
65 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 ...
4
votes
2answers
139 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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votes
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
votes
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 ...
2
votes
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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votes
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
votes
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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votes
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 ...
0
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0answers
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 ...
0
votes
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 ...