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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31 views

find the number of solutions to $p(z) = z^6 + 9z^4+z^3+2z+4$

Let $p(z) = z^6 + 9z^4+z^3+2z+4$ find then number of roots in each quadrant of the complex plane find in which quadrant exists a root which is inside the unit circle using the Argument priniciple ...
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3answers
33 views

Help regarding an exercise in complex analysis

Let $f, g$ be entire functions and $ |f(z)| \le (1+|z|) |g(z)|$ for every $z \in \Bbb C$. Prove that there exist $λ, μ \in \Bbb C$ with $|λ|,|μ| \le 1$ so that $f(z)=(λz + μ)g(z)$. If $ g(a)=0 $ for ...
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1answer
13 views

A set $I$ of isolated complex numbers such that $[0,1]\subset\{Re(z):z\in I\}$

Is there a set $I$ of isolated complex numbers, such that $$[0,1]\subset\{Re(z):z\in I\},$$ where $Re(z)$ is the real part of the complex number $z$.
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0answers
25 views

Visual understanding of radius of convergence for complex power series

I am examining Needham's proof (Visual Complex Analysis 2.III.2) that $\sum c_k z^k$ converges at $a$ implies absolute convergence (and hence convergence) for $|z| < |a|.$ Despite the book's ...
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1answer
24 views

Finding cubed roots of complex number

Is this correct? $a^3 =r^3e^{i3\theta}= 5\sqrt{5}e^{i\arctan(11/2)}$ $$\implies r=\sqrt{5}, 3\theta = \arctan(11/2)+2\pi n,n\in\Bbb Z$$ $$\theta = \frac{\arctan(11/2)+2\pi n}{3}$$ $$\theta = ...
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1answer
17 views

A billinear transformation

Let $$w(z)=(az+b)/(cz+d)$$.Then $w(z)$ maps a straight line of $z-$plane to the circle $|w|=1$ in $w-$plane if $1.|b|=|d|$ $2. |a|=|c|$ $3. |a|=|d|$ $4. |b|=|c|$ My work: I started by considering ...
19
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2answers
537 views

Simpler way to evaluate the Fourier transform of $\exp\left(i e^x\right)$?

I have the task to evaluate $|a(k)|^2$ with $$ a(k) = \int_{-\infty}^\infty \!dx\,\exp\left(i k x + i e^{x}\right).\tag{1}$$ The integral in (1) can be evaluated explicitly via the substitution ...
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1answer
1k views

Straight Line Equation in Complex Plane

Hi there, I'm confused about the straight line equation in complex plane: how does "0 = Re((m+i)z + b)" come from "y = mx + b" ? I mean when I see "y = mx + b", I can draw a graph in my mind, but ...
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1answer
46 views

Branch cut of $\sqrt{z}$

In my complex analysis book, the author defines the $\sqrt{z}$ on the slit plane $\mathbb{C}\setminus (-\infty,0]$. I understand this is done because $z^2$ is not injective on the entire complex ...
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2answers
45 views

Does this set tend towards a disc?

Let $p$ be a complex polynomial \begin{gather*} p:\mathbb{C}\longrightarrow\mathbb{C},\\ \deg p = n,\quad n\in\mathbb{N}. \end{gather*} Define the set $\mathcal{R}=\{z\in\mathbb{C}:|p(z)|\leq R\}$, ...
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2answers
112 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
22 views

Show there's an analytic functions satisfying certain conditions

Suppose that $f$ is analytic on $\{z: |z|<r\}$ for some $r>1$. Suppose further that $|f(z)|<1$ for $|z|=1$ and $f\left(\frac{1}{2}\right) = \frac{1}{2} $. 1) Find a set $U$ so that $f(0) \in ...
2
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1answer
53 views

Compute $\int_0^\infty \frac{x \sin(ax)}{1+x^4} \, dx$

Compute $\displaystyle \int_0^\infty \frac{x \sin(ax)}{1+x^4} \, dx$. Thoughts so far: I see that the function is odd, so is one half the integral on the whole real line. So a half circle contour ...
3
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2answers
42 views

Is the graph of $xy=1$ in $\mathbb C^{2}$ connected?

The graph of $xy=1$ in $\mathbb C^{2}$ is set of points $(x+iy,u+iv)$ that satisfies $$xu-yv=1$$ and $$uy+xv=0$$ How to find if this set is connected or not . I also have another ...
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0answers
19 views

Analysing a set in the complex plane

I would like you to follow my logic, confirming it if correct, suggesting change when flawed or suboptimal. Let $p$ be a complex polynomial \begin{gather*} p:\mathbb{C}\longrightarrow\mathbb{C},\\ ...
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1answer
20 views

Is the magnitude of the gradient non zero?

Let $f=u+iv$ be a holomorphic function on a domain $\Omega$. Suppose $x_{0}+iy_{0}=z_{0}\in\Omega$ such that $f^{\prime}(z_{0})\neq0$ and $\left\vert f(z_{0}) \right\vert > 0$. Let ...
5
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3answers
175 views

All continuous functions are analytic

This might be very silly to ask, but somehow this sequence of results are leading me to this wrong result. I am dealing with complex analysis and the mistake I am making might be because I am using ...
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2answers
505 views

Most general linear transformation which leaves the origin fixed and preserves all distances?

I'm working on the following problem from Ahlfors (complex analysis): Prove that the most general (linear) transformation which leaves the origin fixed and preserves all distances is either a ...
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0answers
12 views

Questions about the inverse of Joukowski function

The inverse of Joukowski function is $ w=z+\sqrt{z^2-1} $. Please show me how to calculate the argument of w and why the branch points of this function are the same as that of $ f(z)=\sqrt{z^2-1} $.
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2answers
21 views

Proof using Identity theorem?

I need to prove that no two distinct holomorphic functions agree on all of $\frac{1}{n}$ where $n$ is an integer. So the identity theorem says that two functions $f,g$ are identical iff the set of ...
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1answer
19 views

Trouble with integration in order to find analytic function

Let $u(x,y) = x/(x^2 - y^2)$ Find $v(x,y)$ such that $f(z) = u + iv$ I'm applying Cauchy-Riemann $u_x = -\frac{(x^2 - y^2)}{(x^2 + y^2)} = v_y$ But I don't see how to integrate that with respect ...
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2answers
29 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
17 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 ...
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3answers
56 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
14 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
419 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
31 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 ...
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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
122 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
50 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
16 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) = ...
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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
15 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
27 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
52 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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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
27 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 ...
0
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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
55 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
64 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 ...
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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
38 views
1
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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 , ...
1
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1answer
36 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 ...