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
60 views

can $f$ be holomorphic?

Let $f:\mathbb{C}\to\mathbb{C}$ be a complex valued function of the form $f(x,y)=u(x,y)+iv(x,y)$. Suppose that $u(x,y)=3x^2y$. Then $f$ cannot be holomorphic on $\mathbb{C}$ for ...
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4answers
154 views

At $z = 0$ the function $f(z)=(e^z+1)/(e^z-1)$ has what type of singularity

At $z = 0$ the function $f(z)=(e^z+1)/(e^z-1)$ has 1. A removable singularity 2. A pole 3. An essential singularity 4. The residue of $f (z )$ at $z = 0$ is $2$. i am completely stuck on it.please ...
3
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1answer
447 views

Question about a problem on the argument principle

Suppose $\Omega$ is a bounded domain in the complex plane whose boundary consists of $m+1$ disjoint analytic simple closed curves. Let $f$ be a non-constant holomorphic function on a neighborhood of ...
2
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1answer
109 views

Showing $F(z)$ is analytic without using the definition of derivative

Here is another problem from Complex Analysis. I think this is the most common question we ever see in exams. If $f$ is continuous on a Jordan arc $\gamma$, prove that the function: ...
2
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1answer
73 views

Limit $\lim_{n\to\infty}\frac{\exp(ia_1)+\exp(ia_2)+…+\exp(ia_n)}{n}=\alpha$

Show that for any sequence $a_1,a_2,...$ of real numbers, the two conditions $\lim_{n\to\infty}\frac{\exp(ia_1)+\exp(ia_2)+...+\exp(ia_n)}{n}=\alpha$ and ...
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1answer
653 views

Accumulation point of zeros of an analytic function

I don't know how to solve the following problem. Any help will be appreciated. Let $f:\mathbb{C}\setminus\{0\}\to\mathbb{C}$ be an analytic function. Suppose $0$ is accumulation point of the zeros of ...
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1answer
409 views

Evaluate a definite integral involving Airy functions

I'd like to show: $$ I \equiv \frac{1}{(2 \pi {\rm i})^2} \int_{a - {\rm i} \infty}^{a + {\rm i} \infty} \int_{b - {\rm i} \infty}^{b + {\rm i} \infty} \frac{1}{{\rm Ai}(u) {\rm Ai}(v) (u-v)} \,dv ...
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2answers
280 views

The preservation of cross ratio

How to prove that if $a,b,c,d$ and $a',b',c',d'$ are 2 quadruples of distinct points in extended complex plane, and if the cross ratios of these quadruples are equal then there exists Möbius ...
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2answers
131 views

Is this an equivalence relation (reflexivity, symmetry, transitivity)

Let $\theta(s):\mathbb{C}\to \mathbb{R}$ be a well defined function. I define the following relation in $\mathbb{C}$. $\forall s,q \in \mathbb{C}: s\mathbin{R}q\iff\theta(s)\ne 0 \pmod {2\pi}$ (and) ...
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1answer
124 views

proving a function is zero everywhere

Given $f$ is holomorphic on the domain $ U := \mathbb{C} \backslash \{0\}$ and that $$|f(z)| \leq |z|^{1/2}$$ for all $z \in U$. How does one prove $f$ is zero everywhere?
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2answers
336 views

Evaluate the following contour integral…

Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t\leq 2\pi$. Evaluate: $$\int_{\gamma(0,1)} \frac{\cos(z)}{z^2}dz$$
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1answer
107 views

Image of horizontal line under Möbius Transformations

What is the image of horizontal line through $i$ under the Möbius tranformation that interchanges $0$ and $1$, and maps $-1$ to $1+i$?
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1answer
735 views

Coefficients of a cubic equation having one positive real root and two complex root with negative real part

Let $0 \lt \alpha \lt 1$ and $\beta,\gamma \gt 0$. Let $p(x) =x^{3}-\gamma x^{2}-\alpha x-\frac{\beta }{\gamma }$. Can we choose $\alpha ,\beta ,\gamma $ such that $p(x)$ has one positive real root ...
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5answers
459 views

Prove that an infinite sequence of nested closed intervals contains a point common to all intervals.

Not a homework question, came across this exercise in Churchill's complex analysis. I haven't done any analysis before so I'm not sure how to answer it. We have a closed interval $$a_0 \leq b_0$$ ...
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2answers
373 views

The Bound of the 8th Derivative of an Analytic Function

This is another question from a recent qualifying exam that really stumped me. I was thinking of using something with the Cauchy estimate for derivatives, but was clueless beyond that. Let ...
2
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0answers
47 views

multiple choice question on holomorphic functions [duplicate]

Possible Duplicate: analytic functions defined on $A\cup D$ Let $f$ and $g$ be holomorphic functions defined on $A\cup D$ , where $A=\{z\in\mathbb{C}:1/2<|z|<1\}$ and ...
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1answer
160 views

Applying the argument principle to functions involving e^z

Typically, you can find the number of zeros of a pole-free function by finding the image of a very large circle in the complex plane. How do you do this if the function includes e^z? On the circle, ...
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2answers
429 views

Prove that it's impossible to approximate $1/z$ with polynomials on an annulus

I heard a nice problem, presumably from an old qual, that I thought I'd share. Problem: Let A be the annulus (in the complex plane) $A=\{z: r_1 \leq |z|\leq r_2\}.$ Prove that $f(z) = 1/z$ cannot be ...
8
votes
1answer
346 views

Polynomial bounded real part of an entire function

Let $f(z)$ be an entire function whose real part is bounded by a polynomial in $|z|$. Does it follow that $f(z)$ is a polynomial? Or, without loss of generality and more suggestively ...
4
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5answers
323 views

Fourier transform of $f(x)=\frac{1}{x^2+6x+13}$

How to find the Fourier transform of the following function: $$f(x)=\frac{1}{x^2+6x+13}$$
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1answer
100 views

Show That This Complex Sum Converges

For complex $z$, show that the sum $$\sum_{n = 1}^{\infty} \frac{z^{n - 1}}{(1 - z^n)(1 - z^{n + 1})}$$ converges to $\frac{1}{(1 - z)^2}$ for $|z| < 1$ and $\frac{1}{z(1 - z)^2}$ for $|z| > ...
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1answer
441 views

Application of Rouche's Theorem

Here is one of the comp questions I need to solve. Find the smallest integer N such that the polynomial $p(z)=2z^5-9z+2012$ has a zero in the open disk of radius N centered at the origin. How many ...
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3answers
189 views

Constant analytic function inside the disk

$f$ is analytic in $\mathbb{D}$ and continuous on $\mathbb{D}$ closure. If $f(e^{i\theta})$ is a real number for $\theta$ in between $0$ to $2\pi$. Prove that $f$ is constant. Also I want to know ...
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1answer
206 views

Normal convergence of sum of analytic function [duplicate]

Possible Duplicate: Uniform convergence of infinite series Suppose that $f(x)$ is analytic in $\{z: |z|<1\}$ and $f(0) =0$. Prove that $\sum f(z^n)$ converges uniformly on compact ...
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1answer
214 views

Evaluating complex integrals involving log (finding bounds)

When evaluating real integrals involving log, I am having trouble with the step that involves finding a bound on circular segments. Let me explain what I mean: If, for example, we have $$ ...
4
votes
2answers
121 views

Sequence of analytic functions

Let $G,H$ be disjoint open subsets of $\mathbb{C}$ and $f_n:G\to H$ be analytic functions. If $f_n(z)\to f(z)$ for all $z\in G$, then prove that $f$ is analytic and $f(G)\subset H$. Any help is ...
1
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1answer
347 views

Non zero analytic functions on annulus

Let $f$ be analytic and nowhere zero on $0<|z|<1$. Prove that $f(z)=z^n \exp(g(z))$ for some integer $n$ and $g$ analytic in $0<|z|<1$.
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1answer
79 views

existence of a certain analytic function

let $f$ to be an analytic function on $D=\{z\in \mathbb{C}: |z|<1\}$. Show that there exists an $\epsilon\in (0,1)$ such that for any natural number $m$, there is an analytic function $g=g_m$ on ...
3
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1answer
261 views

$f$ be analytic on $D=\{z \in \mathbb C: |z|<1\}$ and $f(0)=0.$ Define $g(z)=f(z)/z;z \neq 0$ and $g(z)=f'(0);z=0.$

I was thinking about the following problem: Let $f$ be analytic on $D=\{z \in \mathbb C: |z|<1\}$ and $f(0)=0.$ Define $g(z)=f(z)/z;z \neq 0$ and $g(z)=f'(0);z=0.$ Then which of the following ...
0
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1answer
228 views

compact sets of complex numbers

Let $S$ be a set of complex numbers. I would like to prove the following are equivalent: 1) every sequence of elements of $S$ has a point of accumulation in $S$; 2) every infinite subset of $S$ has ...
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votes
2answers
261 views

Trigonometric integrals over $[0, 2\pi]$

Evaluate the integral $$ \int_0^{2\pi}\frac{1}{3-2\cos \theta +\sin\theta}\,\mathrm d\theta. $$
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2answers
312 views

analytic functions defined on $A\cup D$

Let $f$, $g$ be analytic function defined on $A\cup D$ where $A = \{z \in \mathbb{C}: \frac{1}{2}<|z|<1\}$ and $D = \{z \in \mathbb{C}: |z-2|<1\}$ Which of the following statements are true? ...
1
vote
1answer
325 views

Multiple-choice question about properties of an entire function

Let $f$ be an entire function such that $\lvert f\rvert$ approaches infinity as $\lvert z\rvert$ tends to infinity. Then $f(1/z)$ has an essential singularity at $0$. $f$ cannot be a polynomial. $f$ ...
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2answers
157 views

contour integration along closed curve

suppose $I_r= \int dz/(z(z-1)(z-2))$ along $C_r$, where $C_r = \{z\in\mathbb C : |z|=r\}$, $r>0$. Then $I_r= 2\pi i$ if $r\in (2,3)$ $I_r= 1/2$ if $r\in (0,1)$ $I_r= -2\pi i$ if ...
1
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1answer
141 views

complex mapping

Let $U$ be an open set of $\mathbb{C}$ containing $D=\{z\in \mathbb{C}: |z|<1\}$ and let $f:U\to \mathbb{C}$ be map defined by $f(z)= e^{iψ} \frac{z-a}{1-\overline{a}z}$ for $a\in D$ and ...
1
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0answers
74 views

analytic function [duplicate]

Possible Duplicate: analytic functions defined on $A\cup D$ Let $f$, $g$ be analytic function defined on $A\cup D$ where $A = \{z \in \mathbb{C}: \frac{1}{2}<|z|<1\}$ and $D = \{z ...
3
votes
2answers
810 views

Proving an Entire Function is a Polynomial

I had this question on last semesters qualifying exam in complex analysis, and I've attempted it several times since to little result. Let $f$ be an entire function with $|f(z)|\geq 1$ for all ...
15
votes
3answers
429 views

A question on convergence of series

Suppose $(z_i)$ is a sequence of complex numbers such that $|z_i|\to 0$ strictly decreasing. If $(a_i)$ is a sequence of complex numbers that has the property that for any $n\in\mathbb{N}$ $$ ...
3
votes
1answer
121 views

Branch question

I understand that in complex analysis $\arg(z) = \operatorname{Arg}(z) + 2k\pi i$. In some texts about complex analysis I read things like $\arg_{\tau}(z)$. What does $\tau$ mean? In addition, what ...
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2answers
181 views

The maximum of $|f|+|g|$ is in the boundary

$f$ and $g$ are holomoprphic functions in $G \subset \mathbb C$ and continuous on the boundary of $G$. Prove that $|f| + |g|$ gets its maximum in the boundary of $G$. I know this has something to do ...
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3answers
474 views

Analytic function in the punctured plane satisfying $|f(z)| \leq \sqrt{|z|} + \frac{1}{\sqrt{z}}$ is constant

I saw this question on my book (Complex Analysis/Donald & Newman): Let $f(z)$ be an analytic function in the punctured plane $\{ z \mid z \neq 0 \}$ and assume that $|f(z)| \leq \sqrt{|z|} + ...
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3answers
247 views

Question involving entire functions

Let $f:\mathbb{C}\to\mathbb{C}$ be entire function and $g:\mathbb{C}\to\mathbb{C}$ be $g(z)=f(z)-f(z+1)$. Which of the following statements are true? a. If $f(1/n)=0$ for all positive integers $n$, ...
4
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1answer
355 views

An entire function with two periods

Can anybody help me with this question: If $f(z)$ is an entire periodic function and it has to periods $2$ and $2i$, how can I find all other periods?
9
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2answers
578 views

Zeros of Fourier transform of a function in $C[-1,1]$

I am trying to prove the following: Let $g \in C[-1,1]$. Then the function $$G(z) = \int_{-1}^1 e^{itz}g(t)dt$$ has infinitely many zeros. I know that $G(z)$ is entire and $\lim_{x \to \pm ...
1
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1answer
168 views

complex integration along closed contour

Let $I_r= \int dz/(z(z-1)(z-2))$ along $C_r$, where $C_r = \{z\in\mathbb C : |z|=r\}$, $r>0$. Then a. $I_r= 2\pi i$ if $r\in (2,3)$ b. $I_r= 1/2$ if $r\in (0,1)$ c. ...
5
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2answers
258 views

Intuition behind Cauchy Riemann equations and power series representation

The Cauchy Riemann equations in effect say that a function $f(z) = u(z)+iv(z)$ can be approximated as roughly a scaled rotation $$f(c+h) \approx f(c) + f'(c)h = f(c) + \begin{bmatrix}u_x & ...
1
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1answer
137 views

Boundary of the image of the unit disk under the exponential map.

Anyone know the Cartesian coordinate equation for the top half of the boundary of the image of the unit disk under the exponential map in $\mathbb{C}$? Finding parametric equations in Cartesian ...
6
votes
1answer
166 views

How does one see the topology of a Riemann surface from the graph (assuming one can picture $\mathbb R^4$)?

Given a function $f:\mathbb C\to\mathbb C$ which we will assume is analytic, we have an embedding $f\subseteq\mathbb C\times\mathbb C\cong\mathbb R^4$ of a surface. My question is with regards to how ...
7
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3answers
425 views

Is $\sqrt{z}$ a meromorphic function?

The literature seems rather coy on this point. While $\sqrt{z}$ is not meromorphic on the complex plane $\mathbb{C}$, can it be regarded as globally meromorphic on the appropriate Riemann surface ...
2
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0answers
72 views

asymptotics of $ J_{iu} (ia)$ for a Bessel function

Let $J_{iu}(ia)$ be the Bessel function of imaginary order. ($a$ is a real number (positive or negative) and $u$ is also real.) In the limit $u \to \infty $ how does the function $J_{iu} (ia)$ ...