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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1answer
27 views

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

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

If lim f(z)=0 as z->$z_0$=0 and g(z)<M, with M being a positive number the limit of f(z)g(z)=0

I just wanted to verify my proof here: lim z_>$z_0$ implies that for all $\epsilon.0$ there exists a positive $\delta$ s.t. |f(z)-0|<$\frac{\epsilon}{M+1}$ for all |$z-z_0$|. ...
1
vote
1answer
29 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 ...
1
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0answers
39 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
33 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 ...
0
votes
1answer
14 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 ...
1
vote
2answers
50 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.
0
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1answer
21 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 ...
0
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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
votes
2answers
150 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
votes
0answers
41 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
votes
1answer
23 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
21 views

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

I am unable to find any example of such function?
10
votes
4answers
132 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} ...
1
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2answers
37 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?
0
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0answers
18 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 ...
-2
votes
1answer
34 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 ...
1
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0answers
22 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
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 ...
0
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0answers
22 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
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1answer
62 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
vote
2answers
113 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}} }+ ...
0
votes
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
votes
2answers
54 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 + ...
0
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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 : $\limsup_{x \rightarrow \infty}$ ...
0
votes
1answer
30 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.
7
votes
3answers
222 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.
1
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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} - ...
0
votes
1answer
14 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
49 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)\ \ \ \ ...
0
votes
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 ...
0
votes
1answer
36 views

Regarding Cauchy-Schwarz inequality

I have extracted the following from Ahlfors's Complex Analysis: To prove the Cauchy-Schwarz ...
4
votes
1answer
23 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 ...
1
vote
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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votes
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...
0
votes
1answer
31 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 ...
1
vote
3answers
53 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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votes
0answers
55 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
33 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
votes
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
votes
1answer
47 views
+100

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 ...
1
vote
1answer
21 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
vote
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 ...
0
votes
1answer
48 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 ...
-5
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.