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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26 views

What's “a little computation” here?

I tried to ask the other person in a comment but he's not replied so I'm asking this here: How did he arrive at this formula $$2\pi i f(w) = \int_{\partial D} \frac{f(z)}{z-w}\,dz + \iint_D ...
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2answers
44 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
19 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
11 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 ...
4
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2answers
90 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 ...
2
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1answer
50 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
38 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
19 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
17 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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3answers
108 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
33 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
17 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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votes
1answer
29 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. How one can determine $a, b$ and $p$ such that 1) I ...
1
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0answers
21 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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0answers
16 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
19 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
48 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
39 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
108 views

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}} }+ ...
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2answers
35 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
51 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
26 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
27 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
201 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
34 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 ...
4
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1answer
46 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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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)\ \ \ \ ...
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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 ...
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1answer
13 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
34 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
27 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
15 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
30 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
47 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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0answers
54 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
votes
0answers
32 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
42 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
29 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
20 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 ...
1
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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
45 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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votes
1answer
46 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
18 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
votes
1answer
23 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
votes
0answers
44 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 ...
1
vote
1answer
32 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
vote
1answer
30 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 ...