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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93 views

Solve integrals using residue theorem? [on hold]

$$\int_{0}^{\pi}\frac{d\theta }{2+\cos\theta}$$ $$\int_{0}^{\infty}\frac{x }{(1+x)^6} dx$$ My problem is that I don't know how to start solving these integrals, or how to convert them into usual ...
2
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1answer
34 views

Separate real and imaginary part of $\arccos(z)$

Beginning with $$i \cos \left[ \frac{1}{n} \arccos \left( \frac{i}{\epsilon} \right) + \frac{m \pi}{n} \right]$$ where $m,n \in \mathbf{Z}$, $\epsilon >0$, $\epsilon \in \mathbf{R}$ and $i$ is ...
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0answers
51 views

Using Rouché prove that $a_0\geq a_1 \geq … \geq a_n > 0$ the polynomial $P(z)=a_n z^n + … + a_1 z + a_0$ has no roots $|z|<1$

As asked in the title I am trying to prove this using Rouché's theorem. But I am having a hard time trying to find a second polynomial $f(z)$ such that $$|P(z)-f(z)| < |f(z)|$$ on the unit ...
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0answers
19 views

Total derivative of a complex function using Wirtinger derivatives

The mathworld.wolfram page on Cauchy-Riemann Equations states that given a complex function $f(z) = f(x,y) = u(x,y) + iy(x,y)$ has the derivative: $$\frac {df} {dz} = \frac {\partial f} {\partial x} ...
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3answers
60 views

Complex numbers $z$ satisfying $|z−a|+|z+a| = 2|b|\Leftrightarrow |a|\le |b|$.

could you help me to prove the next statement? Show that there are complex numbers $z$ satisfying $$|z−a|+|z+a| = 2|b| $$ if and only if $|a| \le |b|$. I did the first implication using the ...
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1answer
24 views

convergence radius and sum of a series

1) Given $S(z)$ find the radius of convergence and sum. $$S(z)= \sum_{n \geq 1} \frac{(4z-2)^n}{n}$$ Then: $$S(z)= \sum_{n \geq 1} \frac{(4z-2)^n}{n} = \sum_{n \geq 1} ...
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2answers
39 views

Solve complex exponential equation

I need to solve an expression of this kind (solve for $x$): $e^{\pi i x} -e^{-\pi ix} = 2yi$ Both $x$ and $y$ are real numbers, $y$ is given. I have no clue on how to solve it analytically. All I ...
2
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1answer
58 views

Using the complex logarithm as a conformal mapping,

I want to map the upper half plane, y>0, conformally onto the semi-infinite strip u>0, $-\pi < v < \pi$ in the w-plane. I then studied the complex logarithm, and noticed that the principal ...
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1answer
21 views

Proper definition of concyclic?

Let $z_1,z_2...,z_n$ be points in the complex plane, then if there exists $Z$ such that $$\vert Z-z_k\vert=a\in\{\text{Real Numbers}\}$$for all $k\in \{1, 2, 3...,n\}$, then $z_1,z_2...,z_n$ are ...
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2answers
38 views

Why does this sum equal zero?

Le}t $\gamma$ be a piece-wise, smooth, closed curve. Let $[t_{j+1}, t_{j}]$ be an interval on the curve. Prove, $$\int_{\gamma} z^m dz=0$$ In the proof it states $$\int_{t_{j}}^{t_{j+1}} ...
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1answer
24 views

Meromorphic function with constant magnitude on closed unit disk

I was reading the solution of a problem in the book "Berkeley problems in mathematics", and last part of the proof is unclear to me. It somehow proves using Maximum principle that the rational ...
2
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3answers
53 views

Find the sum of the roots given no multiple roots.

Find the sum of the roots, real and non-real, of the equation $$ x^{2001} + \left( \frac{1}{2} - x \right)^{2001} = 0 $$ given that there are no multiple roots. I am in a weird situation here. ...
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2answers
102 views

Showing properties of a subset of $\mathbb{C}$

Let $\omega(k)=\alpha_{n}k^{n}+\alpha_{n-1}k^{n-1}+\dots+\alpha_{0}$ be a polynomial of degree $n$ on $\mathbb{C}$. Define $D=\{k\in\mathbb{C}:\text{Re}(\omega(k))<0 \}$. How do i show that ...
1
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1answer
41 views

Finding Laurent series of a function $f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$ [closed]

How do i transform this function into Laurent series $$f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$$ where $ \frac{1}{3} < |z| < 1 $.
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1answer
22 views

Show that the inclusion relation of Hardy spaces is proper

The definition of Hardy spaces for the unit disk is here. It is clear that for $0<p<q\le\infty$, $H^q\subseteq H^p$, by Hölder's inequality. I'm asked to show that the inclusion relation is ...
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2answers
31 views

Could someone explain a step in this proof please

Let $u(z)$ be a continuous function in the disc $|z-z_0|<r$ and let $\gamma_{\epsilon}$ be the circle $|z-z_0|=\epsilon$. Prove, $$\lim_{\epsilon \to 0} \frac{1}{2\pi i} \int_{\gamma_{\epsilon}} ...
6
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0answers
121 views

why this three complex are equal (2014 Pan African olympaid problem) [on hold]

EDIT(This following is 2014 Pan African olympiad problem) Let $$H(p,q)=\dfrac{\omega p}{\omega-1+a(\omega p-q)},a>0$$ where $\omega^3=1,\omega\neq 1$,if ...
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1answer
35 views

Residue of $\frac{e^{iz}}{z^2+4z+5}$ [closed]

I need to find the residue of $\dfrac{e^{iz}}{z^2+4z+5}$ at its singular points. How do I do that?
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0answers
36 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
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0answers
29 views

A holomorphic line integral?

Let $f$ be a holomorphic function on a connected open set $U$ which contains the unit disc $D$, and suppose $|f(z)| \geq 1$ for $|z| = 1$. For $w \in D$, the argument principle tells us that ...
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1answer
40 views

Square root of a function

Let $D$ be a circular annulus in $\Bbb C$ with center at $0$. Put $v(z)=z$, for every $z\in D$. Show that $v$ has no square root measurable function. I think if define function $h$ such that ...
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0answers
31 views

When is a line integral of a holomorphic function holomorphic?

Let $U$ be a connected open set, $z_0 \in U$, and $g$ holomorphic on $U$. I know that if $U$ is simply connected, then $$G(z) := \int_{z_0}^z g(w)dw$$ is holomorphic on $U$, and $G'(z) = g(z)$. Here ...
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0answers
36 views

On the definition of modular forms

In many books, I see people defining modular forms to be holomorphic/meromorphic functions in the upper half plane such that it is invariant under the $|_k$ action of the group ...
2
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3answers
75 views

find all $v(x,y)$ so that $f(x+iy)=u(x,y)+iv(x,y)$ is entire

I'm practicing to write down solutions clearly and thoroughly. Is this a proper answer to this exercise? How can it be improved? Let $u(x,y)=x^3-3xy^2$, find all $v(x,y)$ so that ...
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0answers
56 views

Find $\int_0^{2\pi} \frac{d\theta}{2\pi\cos^{2n}(\theta)} \ n=1,2,3,\dots$ via Residue Theorem

So the question is as follows: Use the Residue Theorem to calculate $$\int_0^{2\pi} \frac{1}{2\pi\cos^{2n}(\theta)} d\theta \quad\quad n=1,2,3,\dots.$$ Now I believe the first step would be to use the ...
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0answers
28 views

If $f=u+iv:D\to \Bbb C$ is analytic on a domain D, is then the curves $u(x,y)=c_1$ and $v(x,y)=c_2$ intersect orthogonally?

If $f=u+iv:D\to \Bbb C$ is analytic on a domain D (an open connected subset of $\Bbb C$), is then the curves $u(x,y)=c_1$ and $v(x,y)=c_2$ intersect orthogonally, for any constants $c_1$ and $c_2$? ...
6
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1answer
63 views

Does there exists an entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?

Does there exists a nonconstant entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line? Clearly,$ f(z)=sin(z)$ is an example of an entire function which is ...
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0answers
34 views

Integral curves of complex function components

How to generally find integral curves $F(x,y,c) = 0$ from $u(x, y), v(x, y)$ when $$ x'(t) = u (x, y),\; y'(t) = v (x, y) $$ where $u, v$ are real and imaginary parts functions of a complex ...
3
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2answers
40 views

plot graph of function $f(z)=\frac{1+z}{1-z}$

I am not able to plot graph of function $f(z)=\frac{1+z}{1-z}$. can anyone tell me how to do this without using any software?
3
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2answers
68 views

Proving that the cross ratio is a Möbius transformation

I'm trying to show that given three distinct points $z_1,z_2,z_3\in\mathbb C$, the rational function $$ f(z) = \frac{(z-z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)} = \frac{(z_2 - z_3)z + (z_1z_3 - ...
1
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1answer
21 views

Extending functions on proper subsets of $\mathbb C$ to functions on proper subsets of $S^2$.

There are a number of nice results about extending holomorphic and meromorphic functions from the complex plane $\mathbb C$ to the Riemann sphere $S^2$. See for instance Does entire function extend ...
4
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2answers
49 views

Complex integral with exponential and tangent

Suppose that $k \in \mathbb{R}.$ Evaluate as a function of $k$ the integral $$I(k) : = \int_{-\pi/2}^{\pi/2} e^{i \ k \ \mathrm{tan}(\phi)} d \phi.$$ Any suggestions on how to approach this problem? ...
3
votes
1answer
42 views

Why this map is a mobius transformation

Question: Let $D_2=\bar D(2,1)$ and $D_{-2}=\bar D(-2,1)$ be the closed disks of radius $1$ centered at $z=2$ and $z=-2$ in the complex plane, respectively. Set $X= \mathbb C-\{D_2 \cup D_{-2} \}$, ...
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1answer
80 views

How do I compute the indicator function of an entire function?

Let $F(z)$ be an entire function of finite exponential type. The indicator function of $F$ is defined as ...
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0answers
26 views

Is the Riemann–Liouville fractional derivative holomorphic in order?

If my understanding of complex analysis is correct then the arbitrary order generalization of Cauchy's formula for repeated integration $$(J^\alpha f) ( x ) = { 1 \over \Gamma ( \alpha ) } \int_0^x ...
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0answers
61 views

Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
2
votes
1answer
63 views

What does $\Bbb R/2\pi$ for a set mean?

I simply cannot figure out what this means. I read this on an article about the scalar product of $2\pi$ periodic functions. it says that < f,g > goes from $\Bbb R/2\pi \to \Bbb C$ (complex) Do ...
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1answer
43 views

A certain Complex line integral

In evaluating the line integral of $\frac{dz}{z-2}$ around the circle $|z-1|=5$ , and also around the square with vertices $3+3i ,3-3i,-3+3i,-3-3i$, I obtain zero in both cases however my textbook ...
3
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1answer
54 views

Find the Laurent series about $z=0$

Let $f(z)=\cfrac{e^{-3z}}{z^2(z-2)^2}$, find the Laurent series about $z=0$. On the region $0<|z|<2$, I get $\cfrac{1}{(z-2)^2}=\displaystyle\sum_{n=1}^{\infty}\cfrac{nz^{n-1}}{2^{n+1}}$, ...
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0answers
38 views

set of numbers satisfying a complex exponential equation

Here is the question: Using the principle branch definition of $z^i$ determine the set of all $z\in\mathbb{C}$ for which $(z^i)^2=(z^2)^i$. My ideas: I took the principle branch to be ...
1
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1answer
48 views

Mapping the upper half plane conformally onto a semi-infinite strip,

Map the upper half y>0 of the z-plane conformally onto the semi-infinite strip u>0, $-\pi<v<\pi$ in the w-plane. I would like some hints for now, please. I'm not sure how to even get started ...
2
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2answers
96 views

Is the limit $\lim\limits_{x\to\infty} {i}^{-x}$ equal to $0$, or doesn't exist?

Can someone show me if this limit exists: $$\lim_{x\to \infty} {i}^{-x}=0$$ or it doesn't exist? Here, $i$ is the unit imaginary part. Thank you for any help.
10
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1answer
87 views

Best estimate using Cauchy integral formula: why is a circle the optimal path?

I once encountered this question from Ahlfors' Complex Analysis. An analytic function $f$ has the property that for $|z|<1$, $|f(z)|\leq \frac{1}{1-|z|}$. Find the best estimate of ...
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3answers
52 views

How to remember/rederive the isomorphisms from the half planes to the unit disc

I know that $$z \mapsto \frac{z-i}{z+i}$$ maps the upper half plane to the unit disc, and $$z \mapsto \frac{z-1}{z+1}$$ maps the right half plane. Is there an intuitive way to construct such maps ...
6
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1answer
65 views

When proving that f(z) is a polynomial, is it enough to consider just one point instead of keeping z arbitrary?

I think so - but I'd rather ask the MSE community too. Say I am given the bound |f(z)| < $|z|^3$, and that f is entire. Show f must be a polynomial. I used Cauchy's Integral Formula for ...
2
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1answer
37 views

Slopes of curves from complex derivative [closed]

Show that the slopes of the level curves$$u(x,y)=\text{constant} \ \ \text{and} \ \ v(x,y)=\text{constant}$$ are respectively given by $$\cot(\arg(f'(z))) \ \ \text{and} \ \ -\tan(\arg(f'(z)))$$ If ...
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2answers
122 views

What does “bounded away from zero” actually mean?

For example, is $f(z) = 1/z$, on the set $0<z<1$ "bounded away from zero"?
4
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2answers
62 views

Proof or Counterexample:Is every open connected set $D \subset \mathbb C$ is a domain of holomorphy?

Def: An open set $D \subset \mathbb C^n$ is called a domain of Holomorphy if there exists a holomorphic function $f$ on $D$ such that $f$ cannot be extended to a bigger set. Is every non empty open ...
6
votes
6answers
272 views

Compute definite integral

Question: Compute $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2}dx.$$ Attempt: I've tried various substitutions with no success. Looked for a possible contour integration by converting this into a rational ...
2
votes
1answer
56 views

Existence of unique circle passing through interior points of unit disk meeting the boundary orthogonally

I am a self-studies and this is a hw problem from a complex analysis scourse I've been doing. The problem set pertains to the topic Automorphism Groups and has a high concentration of fractional ...