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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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
13 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
24 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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2answers
28 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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24 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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21 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
47 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
52 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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19 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
33 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
57 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
34 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
31 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
34 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 ...
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1answer
23 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
24 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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22 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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17 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
20 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 ...
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64 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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20 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 ...
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3answers
41 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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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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58 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
44 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 ...
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1answer
15 views

Showing that $\Re z \le |\Re z| \le |z|$ and $\Im z \le |\Im z| \le |z|$

What I'm wanting to show is that $$\Re (z) \le |\Re (z)| \le |z|$$ and also $$\Im(z)\le |\Im(z)| \le |z|$$ So what I've done so far is to consider $$z=x+iy$$ Using the above $z$ I also said that $$\Re ...
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2answers
52 views

Is this identify valid?

$$\sin(t) \dot{}e^{if(t)} = \sin(t+f(t))$$ I've never seen this identify before but it follows directly from the relation between complex exponentials and the trigonometric functions.
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1answer
23 views

Proving that a complex function is not differentiable anywhere

Show from the definition of the derivative that $f(z) = Re(z)$ is not complex differentiable at any point. Easy with the Cauchy-Riemann equations, but I need to do it a different way. Here's my ...
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16 views

Steepest descent from saddle point

I have the function $w(z)=\frac{1}{3}z^3+z$ where $z=x+iy$, i.e. a complex number. I am asked to find the saddle points of this function and then show the paths of steepest descent are ...
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155 views

Expressing a complex function in terms of z

Use the Cauchy-Riemann equations to determine all differentiable functions that satisfy $Re(f(z))=xy$ I think I know how to do this problem. If we let $z=x+iy$, then $f(z)=u(x,y)+iv(x,y)$. We ...
3
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1answer
51 views

False equations with Euler's Identity [duplicate]

What's wrong with the following equations? $$1 = 1^{-i} = (e^{2πi})^{-i} = e^{-i2πi} = e^{2π}$$ My guess would be the third equation, but I can't really tell why... in the first equation, we use the ...
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0answers
49 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
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1answer
27 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
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1answer
22 views

what is true for a meromorphic function with given condition… [on hold]

I am unable to find any example of such function?
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4answers
152 views

How to compute $\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$

Could you explain to me, with details, how to compute this integral, find its principal value? $$\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$$ $f(z) =\frac{\sqrt{z}}{z^2-1} = \frac{z}{z^{1/2} ...
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2answers
39 views

What is unique about the Möbius transform?

... is it the only map to accomplish a transformation in 2D and keep certain characteristics invariant? Which? What else makes it special to be studied so much?
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19 views

To obtain an explicit expression of f1(z) from the Laurent decomposition of f(z)

How do you do to obtain an explicit expression for f1(z) where the function is f(z)=tan z in the annulus {3<|z|<4}? Let f(z)=f0(z) + f1(z) be the laurent decomposition of f(z), so that f0(z) is ...
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1answer
40 views

Divergence and convergence of the integral. [on hold]

I have the following integral, $$I=\int_a^b |x|^{-p} dx$$ where $a<b$ are finite real numbers and $p\leq 0$ is a non-negative real number. If we start solving it we will come up with the ...
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0answers
28 views

Zeta functions and their poles

consider a zeta function as follows $f(x):=\sum\limits_{m=1}^{\infty}\sum\limits_{n=0}^{\infty}\frac{1}{\left(a\cdot m+n+\frac{1}{2}\right)^{2x}}$, for $a>0$ and $\Re(x)>1$. How can I construct ...
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26 views

lacunary series with Jensen's formula

Let $\gamma_{n}\in \mathbb R$ such that $\gamma=\liminf_{|n|\rightarrow \infty} \frac{\lambda_{n}}{n}>0$. We suppose that $\sum_{n}|c_{n}|<\infty$ and $f(t)=\sum_{n\in\mathbb ...
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0answers
23 views

Is there an analytic bounded function on $\omega \subset \mathbb C\setminus]-\infty;0]$ such that $|f(x)|\leq e^{-x^{1/2}}$

Is there an analytic and bounded function on $\Omega \subset \mathbb C\setminus]-\infty;0]$ such that $|f(x)|\leq e^{-x^{1/2}}$; $x\geq 0$ ?
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1answer
67 views

Cauchy integral formula: can it be proved like this?

Consider the Cauchy theorem: Let $D\subset \mathbb C$ be a domain such that $\partial D$ is smooth and $\overline{D}$ is compact. Let $f$ be holomorphic on $D$ and continuous on the closure. Then ...
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40 views

Show that if $\lvert a \rvert \neq 1$, then the equation $\overline{z}^2 = az^2+bz+c$ has only a discrete number of solutions.

I knew the proof for this at some point, but I'm having trouble piecing it back together. At least, I think the proof I'm thinking of was for this result, or a result which implied this result. The ...
1
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2answers
145 views
+50

For what values is my integral diverging or converging?

Is the following integral convergent $$\int_{\gamma}^{+\infty} \left(1-\dfrac{1}{1+sv^{-1}}\right)\left(\frac{1}{\alpha_1}v^{\frac{2}{\alpha_1}-1} \, e^{-\beta\, v^{\frac{1}{\alpha_1}} }+ ...