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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14 views

$S_{1}\iff S_{2}$ in complexe number

Let : $$a_0 , a_1 , a_2 \in \mathbb{C} \text{ and } :b_0 , b_1 , b_2 \in \mathbb{C}$$ : Show the following equivalence : $$\begin{cases} ( 1 + a_0 ) ( 1 + a_1 ) ( 1 + a_2 ) &=& ( 1 + b_0 ) ( ...
0
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4answers
25 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
7 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 ...
1
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2answers
23 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
32 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
14 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
16 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 $\operatorname{argmax}_z f(z,\overline{z})$ Is ...
2
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0answers
17 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 ...
-2
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3answers
44 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
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
18 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
58 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
19 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
37 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
56 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
39 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
51 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
40 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
22 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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0answers
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 ...
8
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2answers
152 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
votes
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 ...
5
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0answers
44 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} ...
0
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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?
10
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4answers
147 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
38 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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0answers
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
37 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
27 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 ...
1
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0answers
22 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 $|f(x)|\leq e^{-x^{1/2}}$; $x\geq 0$ ?
2
votes
1answer
65 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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0answers
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
137 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}} }+ ...
0
votes
2answers
36 views

Is the set of continuous functions from $[0,1] \rightarrow \mathbb{R}$ closed in the same set from $[0,1]$ to $\mathbb{C}$?

Let $X$ be the set of continuous functions from $[0,1]$ to $\mathbb{C}$, equipped with the norm $\|f\| = \int\limits_0^1 |f(x)|dx$, and let $S$ be the subspace of those functions into $\mathbb{R}$. I ...
2
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2answers
55 views

Show $\lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + \lambda_na_n\rvert < 1$ when $\lvert a_i\rvert < 1$ and $\lambda_i\geq 0$

If $\lvert a_i\rvert < 1$, $\lambda_i\geq 0$ for $i = 1,\ldots,n$ and $\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1$, show that $$ \lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + ...
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0answers
27 views

Analytic bounded in the half right plane is identically zero provided that $\limsup_{x \rightarrow \infty}$ $\frac{\log|f(xe^{ni})|}{x}\leq-n$

I have to prove that an analytic bounded function in the right half complex plane $\mathbb C_{+}$ is the null function, that is $f=0$ if we suppose that f verifies : $\limsup_{x \rightarrow \infty}$ ...
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1answer
32 views

Number of zeros of holomorphic function

Let $f(z)=z^{10}+10ze^{z+1}-9$. How to find number of zeros of $f$ in a unit disk ? Probably, I should use Rouche theorem, but I don't know how.
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3answers
233 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
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2answers
35 views

Polar form equations on the unit circle

If $l \in [0, 2 π)$, $k, n \in N$, proof the following equations: $$\mid{e^{i k l/n} - e^{i (k-1) l/n}}\mid = \mid e^{i l/n} - 1\mid$$ and: $$\lim_{n \to \infty} \sum_{k = 1}^n \mid e^{i k l/n} - ...
0
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1answer
15 views

sketching and domain question

Sketch the region in the complex plane given by $|z+i|<2$, with Im$z<0$, and decide whether this region is a domain or not. So correct me if I am wrong but would the combined region be ...
4
votes
1answer
51 views

If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts

If all the roots of a polynomial $P(z)$ have negative real parts, prove that all the roots of the derivative $P'(z)$ also have negative real parts. Could anyone provide a proof for this please?
-1
votes
0answers
28 views

Radius of Convergence of Complex Series [on hold]

Please help me with this question, I've tried using D'Alembert Ratio but I don't understand when theres the complex z involved, thanks. Find the radius of convergence of: $$\begin{align}a)\ \ \ \ ...
0
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0answers
25 views

Complex Logarithm and Principal branches

I've got a question which I'm stuck on. The complex logarithm: (a) Find all possible values of $\log (\sqrt 3 + i)$. So I've found two values to be $\log 2 + i \frac{\pi}{6}$ and $\log 2 - i(11 ...
0
votes
1answer
14 views

radius of convergence of power series by Hadamard formula

Hadamard formula says:for power series $\Sigma _{n=1}^\infty a_nz^n$, if we put:$\lim_{n\rightarrow\infty}sup|a_n|^\frac{1}{n}=\frac{1}{r}$,then the series is divergent for $|z|>r$. but i'm not ...