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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23 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
22 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
12 views

Interesting properties of complex Gateaux derivatives

The complex Gateaux derivatives are for $\psi=\psi_{1}+i\psi_{2}$ and functional F $dF(\psi,\xi)=d_{\psi_{1}}F(\psi_{1},\xi_{1}-id_{\psi_{2}}F(\psi_{2},\xi_{2}$ and ...
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0answers
15 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
24 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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2answers
131 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
28 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} - ...
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1answer
11 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 ...
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30 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?
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22 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. http://i.imgur.com/Qm2OgHQ.png
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0answers
23 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 ...
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1answer
11 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 ...
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1answer
31 views

Regarding Cauchy-Schwarz inequality

I have extracted the following from Ahlfors's Complex Analysis: To prove the Cauchy-Schwarz ...
4
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1answer
22 views

Integral principal value with $\cos$ and $x^2$

Could you tell me how to solve this integral? $$\int_0^{\infty} \frac{\cos x -1}{x^2}dx$$ I think I should focus on this integral $$\int_{\Gamma} \frac{e^{iz}-1}{z^2+ \varepsilon^2}$$ where ...
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1answer
25 views

Integral with denominator raised to n-th power, residues

I don't know how to calculate this integral: $$\int_{-\infty}^{\infty} \frac{d x}{(1+x^2)^{n+1}}$$ If we denote by $\Gamma$ a curve = semicircle centered at $0$ with radius $R$ + segment $[\ R, R]$, ...
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0answers
14 views

Complex Analysis-conformal Mapping [on hold]

I studied complex analysis roudin book and i have a problem with conformal mapping section, in this section there is a theorem that i'll attache in follow, i couldn't find out how it was proved :i ...
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0answers
19 views

Complex analysis proof of regions

how do i prove that the products of regions in C is also a region? Given that the intersection of these regions is not empty. I just know that a region is a non-empty open connected subset of C...
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1answer
28 views

Chain of inequlities in Complex variables

I am having difficulty understanding the following inequalities which is part of a solution to a problem: Suppose $$|z-1|< 1/2$$ $$|z+1|< 5/2$$ and $$|z|> 1/2$$ Also Suppose ...
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2answers
45 views

Fourier series of a periodic odd function

Given $\ f(\theta)=\theta(\pi-\theta)$ is a $2\pi$-periodic odd function on $[0,\pi]$. Compute the Fourier coefficients of $f$, and show that $\ f(\theta)=\frac{8}{\pi} \sum_{\text{$k$ odd} \ \geq 1} ...
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52 views

Complete Proof of Lindelof Theorem [on hold]

enter link description here![enter image description here][2]would you please solve this question for me: why (2) and (3) imply that (5)
4
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0answers
30 views

Obtaining a single-valued branch of $\ln \left( \frac{z-a}{z-b} \right)$ with a branch cut

It is rather easy to see that the function $$f(z) = \ln \left( \frac{z-a}{z-b} \right)$$ has branch points at $z=a$ and $z=b$, My question is why considering a branch cut "connecting" $a$ and $b$ ...
2
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1answer
41 views

Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
3
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0answers
27 views

Moving the integral $Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds$ past Re(s) = 1.

Given the integral $$Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds,$$ I know that the integrand is holomorphic except for simple poles at ...
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1answer
19 views

How do I compute this Milnor number

I need to compute $\mu (x^5+y^5)=5$ on the point $p=(0,0)\in\mathbb{C}^2$. By definition, for $f\in\mathbb{C}[x,y]$, I have $$ \mu(f)=\dim\dfrac{\mathcal{O}_{(0,0)}}{<\dfrac{\partial f}{\partial ...
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1answer
42 views

Roots less than 1 if at least one coefficient is greater than one

I have this doubt. If you have this equation with $\alpha_i \in \mathbb R$ $$P(z)=1-\alpha_{1}z-\alpha_{2}z^{2}- \cdots - \alpha_{p}z^{p}=0$$ I believe that if there exist an $\alpha$ greater or equal ...
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1answer
39 views

The inverse of a bijective holomorphic

Let $U,V$ are open sets in $\mathbf{C}$, if $f:U\to V$ is holomorphic and bijective, then the inverse of $f$ $f^{-1}:V\to U$ is also holomorphic. How can I show that $f^{\prime}(z)\neq 0$ for all ...
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1answer
45 views

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. [on hold]

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. This is a question on complex numbers.
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1answer
14 views

Extending a harmonic function satisfying a growth condition at an isolated singularity

Consider a harmonic function u on the punctured disc $\Delta(\rho)^*:= \{ z\in \mathbb C: 0 <|z|<\rho\}$ with $\lim\limits_{z \rightarrow 0} z*u(z) = 0$. Prove that $u$ can be written in the ...
2
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1answer
20 views

Prove that space of Laurent series is not Hilbert

Let $z_0 \in \mathbb{C}, s>0,T(z_0,s):=\{z\in\mathbb{C}:|z-z_0|=s\}$ and let $V=V(z_0,s)$ be a vector space over field $\mathbb{C}$ of all Laurent series that are uniformly and absolutely ...
3
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0answers
42 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
2
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1answer
29 views

Lagrange's identity in the complex form

I am trying to show Lagrange's identity in the complex form; that is, $$ \Bigl\lvert\sum_{i = 1}^na_ib_i\Bigr\rvert^2 = \sum_{i = 1}^n\lvert a_i\rvert^2\sum_{i = 1}^n\lvert b_i\rvert^2 - \sum_{1\leq ...
1
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1answer
26 views

Injective holomorphic function is conformal(i.e. nonzero derivative)

STATEMENT: If $f:U\rightarrow V$, where $U,V$ are open subsets of $\mathbb{C}$, is holomorphic and injective, then $f'(z)\neq 0$ for all $z\in U$. Proof: We argue by contradiction, and suppose that ...
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2answers
28 views

Flipping sign of $i$s

Why do we flip the signs of all $i$ s in a complex number when we want to take the conjugate of it? I mean, conjugating means making $x + iy$ into $x - iy$, but given a number of the form: $$\frac ...
3
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1answer
69 views

Find all holomorphic diffeomorphisms $f:\mathbb{CP}^1\to\mathbb{CP}^1$

The complex projective line $\mathbb{CP}^1$ is the complex manifold defined by the quotient of $\mathbb{C}^2-\{(0,0)\}$ by the relation $z\sim w$ if $z=\lambda w$ for $\lambda\in\mathbb{C}-\{0\}$. I ...
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4answers
28 views

Proving arg(z/w)=arg(z)-arg(w)

I need to prove that $$arg\left(\frac{z}{w}\right)=arg(z)-arg(w)$$ However, I am a little stuck as to how to go about this. I know the proof for $arg(zw)=arg(z)+arg(w)$ happens by letting ...
3
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0answers
50 views

Complex Analysis (Complex Mapping) stuck on professor's method of simplification in math notes

I'm having an exam this university semester and need some help with my math notes. I've come accross some problems with the section "Complex Mapping." Link to Image of my Notes: Click Me (see first ...
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0answers
27 views

Ordering in the complex plane [on hold]

Explain why there is no ordering ≺ on the complex numbers that satisfies the usual properties of the relation < on the real numbers I just need help starting this problem. I'm not sure what ...
3
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0answers
41 views

Finding Laurent Series of a function

I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
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0answers
21 views

Claim about the z-transform of a discrete function

Claim: $\lim_{k\to\infty} x[k]$ exist and if finite is $X(z)$ the Z-transform of $x[k]$ has no pole in $|z|>1$ and at most 1 pole at $z = 1$ Attempt: \begin{align*} X(z) &= ...
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1answer
17 views

Harmonics conditions for a plucked string

Given a plucked string which is taken on the interval $[0,\pi]$, and it satisfies the wave equation with $c=1$. The initial position of the string is: $\ f(x) = \frac{xh}{p}$ ($0\leq x\leq p$), and $\ ...
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0answers
128 views

How many complex functions reduce to a given x-y function?

A 2D or x-y coordinate function has a complex analog, which is formed by replacing x with with the complex variable z. That function can then be separated into real and imaginary parts. Graphing the ...
6
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1answer
123 views

Can $f(g(x))$ be a polynomial?

Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions. Is it possible that $f(g(x))$ is equal to a polynomial ? edit Some comments : I was thinking about iterations. So for instance ...
2
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1answer
12 views

Determining uniform convergence of complex power series

I'm working on some practice problems for my complex analysis course, and I'm having trouble with uniform convergence. The question asks whether the following series converges uniformly for ...
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0answers
30 views

problem in complex analysis about the modulus [on hold]

Let $z$ and $w$ be any two complex numbers such that $|z|\leq 1, |w| \leq 1$ and $\bar w z$ $\not= 1$. Prove that $$\frac{|(w-z)|}{|(1-\bar w z)|} \leq 1,$$ with equality if and only if $|z| = 1$ or ...
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25 views

Does uniform convergence on $D\subseteq \mathbb{C}$ imply uniform convergence on all subsets of $D$?

Let $f_n:D\rightarrow \mathbb{C}~\forall n\in \mathbb{N}$. If $(f_n)_{n\in\mathbb{N}}$ converges uniformly on $D\subseteq \mathbb{C}$ against $f:D\rightarrow \mathbb{C}$, does $(f_n)_{n\in\mathbb{N}}$ ...
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1answer
62 views

Find roots of $ω^x+(ω^x)^2+1=x$ [on hold]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
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0answers
14 views

Find the equation of an ellipse in a complex plane.

Find an equation for the ellipse that passes through $3+7i$ if its foci at $i$ and $-1$. This is what i have so far. I know that the equation of an ellipse is $|z-p| + |z-q| = c$. so $|z-p| + |z-q| = ...
2
votes
3answers
70 views

Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
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0answers
41 views

Solve equation $ω^x+ω^{2x}+1=x$ [on hold]

We have that to solve $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right.
0
votes
1answer
30 views

Complex Analysis (Limits)

Let $a, b$ be complex numbers. Use the definition of a limit directly (not just the properties of limits) to prove that $$ \lim_{z \to z_0}az + b = az_0 + b. $$ Sorry for the wrong notation, I do ...