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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20 views

An 11-gon with complex numbers

Let $A_1 A_2 \dotsb A_{11}$ be a regular $11$-gon inscribed in a circle of radius $2$. Let $P$ be a point, such that the distance from $P$ to the center of the circle is $3$. Find $[PA_1^2 + PA_2^2 ...
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0answers
15 views

Proof of conformal mappings onto polygons (Stein)

Recently I am reading Stein and Shakarchi's Complex Analysis and I find a great difficulty in understanding the proof of theorem 4.6 in Chapter 8 (p.242-244), which talks about conformal mappings onto ...
3
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3answers
48 views

Geometric interpretation of inverse complex function?

Function $f\colon\mathbb{R}\to\mathbb{R}$ and its inverse $f^{-1}$ are symmetric over line $y=x$. It's easy to imagine inverse of real function, we just have to "flip" the plot over $y=x$. But what ...
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1answer
12 views

Prove there are two points making two regions not conformally equivalent

Show that there exist distinct points $z_0,z_1$ in the open unit disk $D$ so that $D - \{1/2, 1/3\}$ and $ D - \{z_0, z_1\}$ are not conformally equivalent. Thoughts so far: I'm not sure where to go ...
5
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2answers
418 views

Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic

Let $f: \mathbb{C} \to \mathbb{C}$ be a continuous function such that $f^2$ and $f^3$ are both analytic. Prove that $f$ is also analytic. Some ideas: At $z_0$ where $f^2$ is not $0$ , then $f^3$ ...
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2answers
29 views

Recover the holomorphic function from one of its components using Cauchy-Riemann equations

I got two similar questions: Find the holomorphic function $f(x+iy)$ if $\Re(f(x+iy))=x(3-2y)\text{ and }f(i)=2i$ Find the holomorphic function $f(x+iy)$ if $\Im(f(x+iy))=3(x-1)^2y-y^3\text{ and ...
6
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2answers
2k views

Isolated zeros on closure of a domain

Let $f$ be an analytic function on the open unit disk domain $D$. Suppose also that $f$ is bounded. Since $f$ is bounded I believe that $f$ can be continuously extended to the closed unit disk. I ...
28
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1answer
2k views

Cauchy's Integral Formula for Cayley-Hamilton Theorem

I'm just working through Conway's book on complex analysis and I stumbled across this lovely exercise: Use Cauchy's Integral Formula to prove the Cayley-Hamilton Theorem: If $A$ is an $n \times n$ ...
3
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1answer
120 views

Has anyone ever explored $(\sin{x})^x$ , $(\cos{x})^x$, etc?

I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis for this type of equation. It occurred to me ...
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1answer
34 views

If $f$ is non-constant and entire, prove that there exists a $z_0 \in C$ such that $f(z_0)$ is a positive real number

If $f$ is non-constant and entire, prove that there exists a $z_0 \in C$ such that $f(z_0)$ is a positive real number, without appealing to Picard's theorem. The obvious approach to this would be to ...
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1answer
28 views

Show a certain analytic function must exist

Suppose that $f$ is holomorphic on $D - \{0\}$, where $D$ is the open unit disk. Suppose that $f$ has a pole of order one at $0$, with a residue equal to $n$ for some positive integer $n$. Show there ...
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3answers
46 views

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg$

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg.$ My main attempt was using the Cauchy-Riemann equations on the product in this manner but this did not work out. My thinking: ...
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1answer
15 views

Determine all open sets on which $f$ is analytic.

Given $$f(z) = \frac{1}{x} + i(-\frac{1}{y})$$ Determine all open sets in which $f$ is analytic. My attempt: $$f(z) = u(x,y) + iv(x,y)$$ where $u(x,y) = \frac{1}{x}$ and $v(x,y) = ...
0
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1answer
27 views

Prove that a given function is analytic.

We are given the following function : $g(z)=\sqrt{r}e^{i\frac{\theta}{2}}$ , $(r>0,-\pi <\theta <\pi)$ Also , $g(z)$ is analytic in its domain with derivative : ...
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0answers
14 views

Bilinear transformations

I'm having a problem going about these questions Let $T$ be the bilinear transformation that maps $\infty$ to $0$, $\iota$ to $1$ and $2\iota$ to $2$. Derive a formula for $T$. Obtain the image ...
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1answer
26 views

Proving two domains are not conformally equivalent

Let $D$ be the open unit disk. Show that $D - [-1/2, 1/2]$ and $D - \{0\}$ are not conformally equivalent. Thoughts so far: I'm not sure where to begin, but a hint would most helpful to get me ...
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1answer
17 views

Reference about conformal map

I am here because I want to know if someone knows of some good e fast books or references about conformal map . More precisely I need of the propeties of the conformal maps on manifolds with boundary. ...
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3answers
48 views

Is $\sin( | z^{2}| )$ ,where z is complex, analytic?

I know sin is analytic, but I got myself confused in regards to the $| z^{2}| $. I want to say it is since any input sin takes is fine but I feel there's something I missed. Thanks.
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2answers
97 views
+100

Weierstrass's M-test example for uniform convergence and switching Sum and Integral.

How would I go about finding $M_n$ in \begin{equation} \sum_{n=1}^{\infty} \int_{0}^\infty x^{\frac{s}{2}-1}e^{-\pi n^{2}x}dx \end{equation} to show that it is uniformly convergent? UPDATE: Just ...
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0answers
34 views

About Defintion of an Open Mapping

Definition: A function $f$ is said to be an open mapping if the image of every open set in its domain is itself open. So: If we have $f:K \rightarrow \mathbb{C}$, where $K \subseteq D$ , $D$ is the ...
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2answers
26 views

Verifying Cauchy-Riemann equations for $f(z) = \bar{z}^2/z$ at the origin

We're given a two variable function as follows : $$ f(z) = \begin{cases} \dfrac{\bar{z}^{2}}{z} , & z\neq0 \\ 0\:\:\:, & z=0 \\ \end{cases} $$ We need to show that the ...
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1answer
20 views

Evaluating This Complex Line integral

I'm trying to evaluate the following: $$\int_{\mathcal{C}}z^3 e^{-z^4}\,dz $$ along the path $\mathcal{C}=\left\{\sin(t^2)-i\frac{2t^2}{\pi}:0\leq t\leq\sqrt{\frac{\pi}{2}}\right\}.$ I tried using ...
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1answer
17 views

Find a domain (open and connected set) in which $f(z) = (z-2)arg_0 (z)$ is continuous.

Find a domain (open and connected set) in which $f(z) = (z-2)arg_0 (z)$ is continuous. Note: $$arg_\phi(z) = arg(\phi),~~~~ \text{where }\phi < arg(z) \le \phi + 2\pi$$ \begin{align}f(z) ...
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1answer
54 views

Are all derivatives of sinc function bounded on real axis?

It seems that all derivatives of $sinc$ function ($sinc(x)=sin(x)/x$) are bounded on real axis. Is it true or no?
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1answer
62 views

prove that if $ h = |f_1|^2 \cdots + |f_n|^2$ is constant then $f_i$ is constant

Let G be a domain, and let $f_1 \ldots f_n$ analytics in G such that $$ h = |f_1|^2 + \ldots + |f_n|^2$$ is constant prove that every $f_i$ is also constant in G the question has a hint to ...
1
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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 ...
1
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1answer
16 views

Lipshitz constant of bounded function on disc

Let $f(z)$ be analytic in the unit disc $\{z\in\mathbf{C}:|z|<1\}$ and have bounded modulus in the sense that $|f(z)|\leq M$ for all $z$ in the disc. Let $0<r<1$. Find a constant $C$ ...
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0answers
37 views

How can I prove $e^{i\theta}=\cos\theta+i\sin\theta$? [duplicate]

How can I prove $$e^{i\theta}=\cos\theta+i\sin\theta$$
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1answer
34 views

Find the number of elements of a complex subset

How many elements does the set $\{z\in \mathbb C:z^{60}=-1,z^k\not=-1\text{ for } 0<k<60\}$ have ? $z^{60}=-1=\cos(2k\pi+\pi)+i\sin(2k\pi+\pi)$. Then , ...
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1answer
35 views

An alternative method to find $\sum_{k=1}^{2n-1} | \beta ^k - 1|$

Let $\beta \in \mathbb{C}$ such that $\beta ^n = 1$ but $ \beta ^k \neq 1$, $\forall k=1,2,\cdots, n-1$. Find the value of $$ \sum_{k=1}^{2n-1} | \beta ^k - 1|.$$ I came across such a question ...
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0answers
10 views

a,b and c are on complex plane are on the unit circle [on hold]

Suppose that a, b and c are on the unit circle in the complex plane and a + b + c = 0. Prove that a, b and c are the vertices of an equilateral triangle. Find the general expression ...
0
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1answer
54 views

Show that there exists holomorphic $f$ such that $f^2=\frac{\sin z}{z}$

I have to show that there exists a holomorphic function $f$ on a neighborhood of $0$ such that $f^2(z)=\frac{\sin z}{z}$ on this neighborhood. Furthermore, I have to find the radius of convergence of ...
3
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2answers
27 views

Find the image of $|z+1|=2$ under $f(z) = \frac{1}{z}$ where $z \in \mathbb C$

Find the image of $|z+1|=2$ under $f(z) = \frac{1}{z}$ where $z \in \mathbb C$ My attempt: Let $z = x + iy$ $\displaystyle |z+1|=2 \iff | (x + iy)+1|=2 \iff |(x+1) +iy|=2 \iff (x+1)^2 + y^2 = ...
2
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1answer
53 views

find all zeroes of $p(z) = \sum_{k=0}^n \frac{z^k}{k!}$

show that all the zeroes of $$p(z) = \sum_{k=0}^n \frac{z^k}{k!}$$ where z is a complex number $z\in \Bbb{C}$ are inside the ring $$ \{\frac{n}{2e} < |z| <2n\}$$
2
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1answer
21 views

Limit of a sequence of holomorphic functions

Let $f_n$ be a sequence of holomorphic functions on a domain $D \subset \mathbb{C}$ converging to a function $f$, and also converging uniformly on compact subsets. Suppose each function has at most ...
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1answer
19 views

Is there a holomorphic function on the unit disk that satisfies a certain condition?

Is there a holomorphic function on the unit disk that satisfies $f(1/n) = 1/\sqrt{n}$? Thougthts so far: I know that $f(z) = \sqrt{z}$ won't work, as it is not analytic at $0$. My intuition says that ...
2
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1answer
75 views
+50

Showing existences of biholomorphic maps.

Define a complex polynomial $p:\mathbb{C}\longrightarrow\mathbb{C}$ where $\deg p=n\in\mathbb{N}\setminus\{2\}$. Define the set $\mathcal{S}=\{z\in\mathbb{C}:\text{Re }p(z) < ...
3
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1answer
25 views

Primitive of $dz/z$ is a branch of log

Let $D$ a connected open set of $\mathbb{C}$. A continuous function $f:D\to \mathbb{C}$ is a branch of log if $e^{f(t)}=t$ on $D$. In my book (Cartan) it is written that if $F$ is a primitive of the ...
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1answer
35 views

Is it necessary for the Imaginary-axis to be perpendicular to the Real-axis?

The Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...
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1answer
11 views

Uniform convergence on punctured disc => analytic

Suppose $f(z)$ is analytic on the punctured disc $D'=\{z\in\mathbf{C}:0<|z|<1\}$. Suppose further that there are functions $\{f_n(z)\}_{n=1}^\infty$ that are analytic on all of ...
2
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1answer
42 views

How to determinate poles and residues of this function.

I have the following function $$f(z)=\frac{e^{iz}}{e^z+e^{-z}}$$ Using the notaion $z= \rho(cos\theta+isen\theta)$ I found that the poles are $z_1=\frac{\pi}2i$ and $z_2=-\frac{\pi}2i$. To determinate ...
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2answers
29 views

Maximal Modulus Principle Theorem for a Compact Set

The theorem says: The maximum of the absolute value of an analytic function over some compact set in a domain of analyticity is attained at the boundary of that compact set. Could someone please ...
0
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2answers
31 views

How to write $e^{2iz}$ in the form $u(x,y)+iv(x,y)$?

I took an exam on Complex Analysis recently, and questions involving the complex logarithm and exponential were a sticking point for me. Questions such as: Q. The function $f$ is defined by $f(z) = ...
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0answers
17 views

How to choose grid for a numerical integral of complex function?

I need to numerically integrate a complex function $f(x)$ on R, i.e. to approximate $\int_{-\infty}^\infty{f(\xi)d\xi}$. Performance is crucial as the integration is repeated a high number of times ...
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1answer
39 views

Set on which a holomorphic function is bounded is simply connected

I came across this problem and couldn't figure it out: Let $U$ be a simply connected domain, and let $f:U\to\Bbb{C}$ be holomorphic on $U$. For $c>0$ define $V_c:=\{z:|f(z)|<c\}$. Show that ...
0
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1answer
43 views

How Can I Proof $1+2\cos(\theta)+2\cos(2\theta)+2\cos(3\theta)+.. $ [duplicate]

Proof $1+2\cos(\theta)+2\cos(2\theta)+2\cos(3\theta)+...+2\cos((n-1)\theta)=\frac{2\sin((n-\frac{1}{2})\theta)}{2\sin(\frac{1}{2}\theta)} $
0
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2answers
24 views

calculating complex numbers - help needed [on hold]

Probably it is simple, but I am blind right now and I do not see how to solve this task: $e^{i \frac{2\pi}{3}}+e^{i\frac{4\pi}{3}}+1$
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0answers
13 views

An application of Rouche. when will roots lie in the annulus

Suppose $0<\rho<1$. Show that for $n\geq N(\rho)$, $nz^{n+1}-(n+1)z^n+1$ has all the zeros in $\rho<|z|<1/\rho$. Provide an estimate of $N(\rho)$. We wish to show that on $|z|=1/\rho$, ...
0
votes
0answers
49 views

$S_n\to S$ implies $\sigma_n\to S$ [duplicate]

Let $(a_n)_{n\ge0}$ be a sequence of complex numbers and $S_n=\sum_{k=0}^na_k$, set $\sigma_n:=\frac{S_0+S_1+\dots+S_n}{n+1}$, if $\lim_{n\to\infty}S_n=S$ then show that ...
12
votes
1answer
161 views

Why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?

Reading through Titchmarsh's book on the Riemann zeta function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = ...