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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Points on a straight line (Complex Analysis)

I encouter a problem in complex analysis course : Let $a, b, $ and $c$ be three distinct points on a straight line with $b$ between $a$ and $c$. Show that $\frac{a-b}{c-b} \in \mathbb{R}_{<0}$. ...
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12 views

Cauchy-Riemann Equations Written as Complex Conjugate

Apparently, it can be shown that the Cauchy-Riemann equations can be written simply as, $df/dz^*=0$. I do not understand how it does not immediately follow from this that $df/dz=0$. When we proved ...
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9 views

Continuity of a function on the ring of formal power series, with a metric defined.

Let $E$ be the ring of formal power series over a field $K$. Consider $S,\ T \in E$. Define a metric $d$ on $E$ by $d(S,T)=0\ $ if $S=T$ and $a^{(-k)}$ for $k=\mathrm{order}(S-T)$, where $a>1$ is ...
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2answers
17 views

An explanation of how the range of a complex function works? Specifically $f(z)=z^2$ for$Re(z)>0$, $Im(z)>0$ and in the first quadrant..

I'm going through this complex analysis textbook, and it tells me that the range of the aforementioned function is $Im(w) \geq 0$. To me, that makes no sense. Could someone explain that, by chance? ...
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16 views

sketching lines and curves in the complex plane.

Well, my question is I a have the line equation x=1 and I need to know wich is thw image under $w=z^{2}$, then I parametrize it like $\theta=\pi /2$, the next step was to squred it $(\theta-\pi ...
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1answer
11 views

Series Expanding a Function: Complex Answer?

I have $$ f(x)=2\arccos\left(\frac{x}{2}\right)-x\sqrt{1-\frac{x^2}{4}} $$ and my friend says that the series of this function about $x=2$ (truncated to the first term) given $x\leq2$ is ...
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0answers
15 views

Coefficient in Taylor Series expansion [duplicate]

Find the coefficient of $(z-\pi)^2$ in the Taylor series expansion around $\pi$ if $$f(z) = \begin{cases} \frac{\sin z}{z-\pi} & \quad, z \neq \pi \\ -1 & \quad, z=\pi \end{cases}$$
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0answers
8 views

Power series of dependent and independent variables

Let $f(z,w)$ be an analytic function in two variables where $w=w(z)$ is dependent on $z$ ($z$ is the independent variable). Then $f(z,w)$ has a power series expansion centered at $(z_0,w(z_0))$ ...
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0answers
19 views

How to define Square Root

I'm trying to understand how to define the square root of a complex function "globally". Let's say we have some function from some set $X$ onto $\mathbb{C} - \{0\}$: $$ f:X\to\mathbb{C}-\{0\} $$ and ...
2
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3answers
47 views

Is the function complex differentiable at (0,0)?

(in Complex) $$ g(z) = \begin{cases} \frac{z^5}{|z|^4} & \text{if $z \neq 0$} \\ 0, & \text{if $z = 0$ } \end{cases} $$ For the function above, is it differentiable at $z=0$? I am trying to ...
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11 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
2
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1answer
35 views

Sketching curves in the complex plane

Well I really need your help here because I need to sketch the curve $|z-1|=1$ in the z-plane and then its image under the $w=z^{2}$ but the thing is that I dont know how to sketch that function. In ...
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0answers
9 views

Integrating around simple pole and semicircle

Let $f$ be a holomorphic function on $\mathbb{C}$ with simple pole at $z_0$. Then if $\Gamma$ is a circle around $z_0$ oriented counter-clockwise with radius $r$ and $r\rightarrow 0$, then ...
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1answer
28 views

Prove for some $z_0 \in C$ the function $f(z)=|z-z_0|$ is continuous on all of $\mathbb{C}$

Let $z_0\in\mathbb{C}$ and $f(z)=|z-z_0|$. Show that $f$ is continuous on $\mathbb{C}$. I expect to see a proof using the triangle inequality. Note a function $f$ is continuous on $\mathbb{C}$ if ...
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0answers
17 views

contour integral and limit: What is the condition of the interchange the order?

In real real analysis sense, the interchange between limit and integral is hold when integrand is uniformly converges. $i.e$ \begin{align} \int \lim f = \lim \int f \end{align} Here i want to ...
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2answers
29 views

Reflections of circles through a circle are circles

To make things easier, we will try to reflect some general circle through the unit circle. We can use the inverse of the Cayley transform to map our analytic arc in the $z$-plane to the real line in ...
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0answers
27 views

power series of $\frac{1}{1-z}$ at $ a$

Here is a problem and I'm confused whether I'm doing correct or not. Problem says, Write down a convergent power series at $a \neq 1$ which represents the function $\frac{1}{1-z}$ and find radius of ...
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1answer
11 views

Conformal transformation of complement of disk in upper half plane

Let $U$ be the complement in the half-plane $\operatorname{Im} z > 0$ of a disk of radius $a<1$ centered at $i$. I am looking for a conformal transformation that maps $U$ onto an annulus. Since ...
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3answers
33 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
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21 views

Cauchy's integral theorem and domain boundaries

On a homework assignment, I was asked if the following statement is true. If $f(z)$ is analytic in a simply connected domain $D$ and continuous in $\partial D$ then $\oint_{\partial D} f(z) = 0$. Is ...
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1answer
18 views

Contour Integration where Contour contains singularity

There are many theorems in complex analysis which tell us about integration $\int_{\gamma} f$ where $f$ is continuous (or even differentiable) in the interior of $\gamma$ except finitely many points. ...
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1answer
27 views

Simple complex analysis inverse

On page 113 of Churchill in explaining the $\arcsin{(-i)}$ it comes across $$ln(1-\sqrt{2})$$ which is fine but then it goes on to say that it is equal to $$ln{\frac{1}{1+\sqrt{2}}}$$ How do they ...
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35 views

Complex differentiability of $f(z)=|z|$

Why is the absolute value function $f : \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = |z|$ not complex differentiable at any point $z_0$ in the plane?
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1answer
27 views

Complex variables Open ball [on hold]

Let $f(z) = \frac1z$ be inversion. Given a real number $a$, let $R_a = \{z \in C : Im(z) < a\}$. Why is $f(R_a)$ an open disk, provided $a < 0$. What happens when $a \ge 0$?
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2answers
24 views

The Duplication Formula for the Gamma Function by logarithmic derivatives.

I was reading Ahlfors' "Complex Analysis" (second edition) and in Chapter 5, section 2.4, where he studies the Gamma Function, he proves Legendre's Duplication Formula: ...
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0answers
49 views

$\epsilon$-$\delta$ proof of a sinc limit in Complex variables [on hold]

I am stuck on the following problem : Prove (using $\epsilon$-$\delta$) that $$\lim_{z \rightarrow \pi/2} \frac{\sin z}{z} = 2/\pi$$ Basically I do not know how to get an estimate on ...
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3answers
49 views

taylor of $\frac{1}{z}$ at $a=-2$

I want to find the taylor series representation of $f(z)=\frac{1}{z}$ at $a=-2$. The point of this exercise is not to find some pattern in the derivatives, infact we are not meant to find any ...
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2answers
60 views

Inequalities involving the sine of Complex Variable z

Is there any relationship between $|\sin z|$ , $\sin |z|$ , and $|z|$ ??? I know in real variables for example we have that $|\sin x|\le|x|$
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1answer
29 views

Relationship of basis vectors of the complex plane

I am working on learning more about the connection of complex numbers and rotations in the context of rational geometry. Thanks ahead of time for any corrections on my best assertions. Let $B$ ...
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2answers
34 views

calculate $\int_{0}^{2\pi}\frac{1-\sin(t)}{2-\cos(t)}dt$

I need to calculate $\int_{\gamma} \frac{1-\sin(z)}{2-\cos (z)}dz$ where $\gamma$ is the upper hemisphere of the circle with center $\pi$ and radius $\pi$, with a positive direction. The original ...
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1answer
60 views

$S_{1}\iff S_{2}$ in complex numbers

Let : $a_0 , a_1 , a_2 , b_0 , b_1 , b_2 \in \mathbb{C} $ : Show the following equivalence : $$\begin{cases} ( 1 + a_0 ) ( 1 + a_1 ) ( 1 + a_2 ) &=& ( 1 + b_0 ) ( 1 + j b_0 ) ( 1 + j^2 b_0 ) ...
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4answers
35 views

Sum of nth roots of unity

Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+...+c^{n-1} = 0$ Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} ...
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1answer
12 views

A question about analytic functions on the unit disk with $\Re[h(z)]=0$ and a double pole at $1$

Let $\bar{D}(0,1)$ denote the closed unit disk around $0$ and $D$ the unit circle. I am interested in obtaining a complex function, say h, with the following properties: $h(z)$ is analytic for ...
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2answers
32 views

Existence of a non-constant entire function

Which of the following statements are true? a. There exits a non-constant entire function which is bounded on the upper half plane $$H=\{z\in \mathbb C:Im(z)>0\}$$ b. There exits a ...
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2answers
40 views

What is the coefficient of $(z-\pi)^2$ in Taylor series expansion of $\sin (z)/ (z-\pi)$

I want to determine the coefficient of $(z- \pi)^2$ in Taylor series expansion of $f(z)=\sin (z)/ (z-\pi)$ if $z \neq \pi $, $-1$ if $z=\pi$ around $\pi$. How can this be done? I don't know how to do ...
1
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1answer
28 views

Conformal mapping of a semi-circle and a finite line

Can I map a semi-circle and a finite line separated by a distance $h$ to two parallel lines? Since I am new to con-formal mapping, I used the $w=atan(z$) con formal function but I guess this is for ...
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0answers
27 views

Find maximum of a function of a complex argument

I'd like to find the maximum of a (real) function of a complex argument. However, the function contains the $\Re(z)$ operator, so that the question is $$\operatorname{argmax}_z f(z,\overline{z})$$ ...
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0answers
23 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
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3answers
48 views

Imaginary part of $ln(\sqrt{i})?$ [on hold]

Which of the following is the imaginary part of a possible value of $\ln(\sqrt{i})?$ (a) $\pi$ (b) $\pi/2$ (c) $\pi/4$ (d) $\pi/8$ I compute $\sqrt{i}=\dfrac{1+i}{\sqrt{2}}$, but how to proceed ...
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0answers
18 views

Infinite product representation for the Sine Integral $\mathrm{Si}(z)$

The infinite series representation of the sine integral (http://en.wikipedia.org/wiki/Trigonometric_integral, previous m.se question: Is there any infinite series representation of the sine ...
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1answer
22 views

Complex var. integral: $\oint_{|z|=1} \frac{z^2\ dz}{\sin^3{z}\cos{z}}$

Integrate $\displaystyle\oint_C \dfrac{z^2\ dz}{\sin^3{z}\cos{z}}$; $C \rightarrow |z|=1$ I already know that $|z|=1$ is a circumference with $r=1$ and center at $(0,0)$. I also know there are ...
3
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0answers
65 views

Infinite sum of analytic function still analytic

Consider $$ f_n(x) = n e^{-n^6(x-n)^2} : \mathbb R \rightarrow \mathbb R$$ and the series $$ f(x) = \sum_{n=1}^{\infty} f_n(x). $$ Is $f$ analytic on $\mathbb R$? A function is analytic if for ...
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0answers
21 views

Complex Mapping of $\mathrm{cosh}(w)=z$

Mapping in complex analysis has not been very easy for me unfortunately. I am having difficult trying to find the mapping between the z and w plane. I attempted to simply write that ...
3
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3answers
44 views

Complex hyperbolic Trigonometry

When faced with the equation $\cos{z}=\sqrt{2}$ I want to solve for z so I break it up into a sum $z=x+iy$ and get: $\cos{z}=\cos{x}\cosh{y}-i \sin{x} \sinh{y}$ equating real and imaginary parts I ...
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0answers
6 views

What is the region ( area) of integration in Double mellin Barnes integral?

What is the region ( area) of integration in Double mellin Barnes integral ? In H-function of two variables we are using double Mellin-Barnes contour integration on s and t planes where s and t are ...
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1answer
59 views

Complex transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ [on hold]

Under the transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ is transformed to (a) $\{z\in \mathbb C:0<\operatorname{arg}(z)<\pi\}$ (b) $\{z\in \mathbb ...
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1answer
46 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
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1answer
33 views

Proving Fundamental Theorem of Algebra using Maximum Principle

I'm trying to prove FTA by using the maximum principle. Here's what I did, Let $P$ be a polynomial of degree at least $1$ and assume that $P$ has no zeros. Define $$f(z):=\frac{1}{P(z)}.$$ Then ...
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0answers
52 views

Is my understanding of the argument correct?

I worked through a proof of: $$ f(z) = {1\over 2 \pi i}\int_{\partial D} {f(w) \over w -z} dw$$ where $D\subset \mathbb C$ is an open disk and $f$ is holomorphic on $D$ and continuous on ...
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3answers
44 views

Geometric proof of complex number equation

Use geometric reasoning to find a value for $θ \in [−\pi, 0]$ satisfying $|e^{iθ} − 1| =\sqrt2$. So far I have converted to exponential form as $|\cos \theta + i\sin \theta -1|=\sqrt2$. I'm having ...