Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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4 views

Find if $W=\lbrace z\in \mathbb{C}\mid 2 \leq |z| \leq 3 \rbrace$ set is connected and compact

Find if $W=\lbrace z\in \mathbb{C}\mid 2 \leq |z| \leq 3 \rbrace$ set is connected and compact. Let $z\in \mathbb{C}$, then $z=x+iy$ for $x,y\in \mathbb{R}$ Notice that $$W=\lbrace z\in \mathbb{C} \...
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0answers
9 views

Complex function $Log(\frac{z-a}{z-b})$ and cauchy-riemann-equation

I need a proof that the following complex function satisfies the Cauchy-riemann equations $$\operatorname{Log}\left(\frac{z-a}{z-b}\right),$$ where $z\in\mathbb{C}$ and $a,b\in\mathbb{R}$ such that $a&...
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1answer
31 views

For what $z$ does the sequence $z_n=nz^n$ converge?

For what $z$ does the sequence $z_n=nz^n$ converge? Attempt Consider $\sum_{n\geq 1}nz^n $ and notice $$\lim_{n \to \infty}\frac{z_{n+1}}{z_n}=(1+\frac{1}{n})z=z$$ but the serie converges iff $|z|<...
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15 views

Why does the inverse Fourier transform differs from the Laplace inverse Bromwich integral?

This might be a repeated question, but I am looking for a more in depth explanation for the relation between inverse Fourier and Laplace transforms. We all know that the inverse Laplace transform is \...
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0answers
33 views

Show that the function $\log\frac{z-a}{z-b}$, with $a<b$ is not an analytic function

If $z=x+iy$ is a complex number show that the function $\log\frac{z-a}{z-b}$, with $a<b$ is not an analytic function in the points $z=x+iy$ with $y=0$ and $a \leq x \leq b$.
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18 views

Cauchy's integral formula and Residue theorem

I am confused when to use Cauchy's integral formula and when to use thee residue theorem. I've seen example that when $f(z)=\frac{2z+1}{(z-2)(z+1)}$ over a contour , you can only use Residue theorem, ...
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7 views

Conformal equivalence to upper half plane

I'm trying to construct a conformal equivalence between the region(inside the disk $D(1,1)$ and outisde the disk $\overline{D(\frac{1}{2},\frac{1}{2})}$) and the upper half plane. How do I construct ...
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1answer
19 views

prove that $\frac{|z-a|}{|\overline{a}z-1|}$ if |z|=1 and a $\in \mathbb{C}$

I started with that: $\frac{|z-a|}{|\overline{a}z-1|}=\frac{|(z-a)(\overline{\overline{a}z} -1)|}{|\overline{a}z-1|^2}=\frac{|a|z|^2 - z -a^2\overline{z} +a|}{|\overline{a}z-1|^2}= \frac{|a^2\overline{...
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4answers
29 views

When $\frac{z-1}{z+1}$ is a real number?

I wrote it this way: $\frac{z-1}{z+1} = \frac{(z-1)(\overline{z}+1)}{(z+1)(\overline{z}+1)}=\frac{|z|^2 + z - \overline{z} -1}{|z+1|^2} = \frac{|z|^2 -1+2iImz}{|z+1|^2}$ and I calculated that for $|z|=...
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1answer
21 views

Contour integral of $f(z)=z$ from $z_1 = 5i$ to $z_2 = 2-2i$

I have to make two different contour integrals of $f(z)=z$ from $z_1 = 5i$ to $z_2 = 2-2i$. I did the first contour to be $\gamma_1 (t) = 5i + t(2-2i)$ with $0\le t \leq 1$. But my question is how can ...
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31 views

Riemann Xi Function $\xi(s)$

If $\xi(\sigma+it_0)$ denotes the Riemann Xi function for fixed $t_0$, If $\sigma>1/2$ then, Prove that $$\Re( \xi(\sigma+it_0) \xi'(\sigma-it_0) )>0 $$ where $\xi'(\sigma-it_0) $ denotes ...
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39 views

Prove that $\sum_{n=-\infty}^{\infty}\frac{1}{(x+\pi n)^2}=\frac{1}{\sin^2x}$ [closed]

$$\sum_{n=-\infty}^{\infty}\frac{1}{(x+\pi n)^2}=\frac{1}{\sin^2x}$$ Marko Riedel Dec 30 '18 at 14:46 Show 5 more comments 1 Answer order by votes Up vote 3 Down vote Accepted With the quoted proof ...
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12 views

Determining that the singularity of a differential equation is a pole only.

The equation is: $$ \frac{\mathrm{d}w}{\mathrm{d}z} = w-w^2 $$ where $w(z)$ is a complex function. To solve, I separated variables and then used partial fraction decomposition to obtain: $$ \frac{\...
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0answers
32 views

Applying the Schwarz Lemma

https://en.wikipedia.org/wiki/Schwarz_lemma Suppose $f$ is a holomorphic function on the complex plane that maps the unit disc $\mathbb D$ into itself, such that $f(0)=0$. Suppose also that $f$ is not ...
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57 views

$\zeta( \frac{z}{z-1} )$ has infinitely many zeros when |z|=1.

Evaluate $$ \oint_{\left\vert z\right\vert\ =\ 1} \log\left(\left\vert \zeta\left(\frac{z}{z - 1}\right)\right\vert\right) \,{\mathrm{d}z \over z} $$ where $\zeta(s)$ is the analytic continuation of ...
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44 views

Jensen's Formula.

Suppose that $f$ is an analytic function in a region in the complex plane which contains the closed disk $D=\overline{B(0,r)}$ about the origin, $a_1, a_2, \cdots, a_n$ are the zeros of $f$ in the ...
3
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2answers
79 views

How to find the sum of $\sum_{n=0}^{\infty} \frac{z^{3 n}}{(3 n) !}$?

I'm really stuck with this problem and I hope some of you could give me a hint. Consider the functions: $$f(z)=\sum_{n=0}^{\infty} \frac{z^{3 n}}{(3 n) !}, \quad f^{\prime}(z)=\sum_{n=0}^{\infty} \...
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0answers
4 views

Bounding the modulus of a complex function which is the amplification factor of a numerical scheme

In my studies of numerical PDEs, I am trying to perform a von Neuman stability analysis for the Lax-Wendroff scheme. I think I have the following amplification factor $$ G(\xi) = \frac{1}{2}\nu (1+\...
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15 views

Convergence of contour integral

I am looking at an answer to this question, and I kind of don't understand how this contour integral works. It makes sense to me that we can evaluate it at two lines, one slightly above the real axis ...
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0answers
42 views

Where is $\varphi(s)=\Gamma\left(1+\frac{1}{s}\right)+\sum_{n\ge0} \frac{(-1)^n}{n!}\zeta(-ns)$ equal to zero?

It's known that when you analytically continue the Riemann zeta function to the complex plane, it has roots. I would like to figure out if there are roots to the following analytic continuation of $\...
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0answers
35 views

Find all meromorphic functions satisfying

Find all meromorphic functions g satisfying the condition that for all $z \in \mathbb{C}$ $|g(z)| \leq K|z+1|^{\frac{-3}{2}} +C |z|^{\frac{3}{2}}$ , ($K$ and $C$ are real constants) what's throwing me ...
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1answer
22 views

Finding the limit of the complex function

I am asked to find the limit $$\lim_{z \to \infty_{\mathbb{C}}} \frac{x^2y-x-2}{1-x}.$$ I have no clue how to solve this because we never covered limits like these?
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1answer
58 views

Evaluate$ \oint _{|z|=1} \frac{\log\ |1-z|}{z}dz $

Evaluate $$ \oint _{|z|=1} \frac{\log\ |1-z|}{z}dz $$ My Attempt $$ I=\oint _{|z|=1} \frac{\log\ |1-z|}{z}dz $$ $$z=e^{i\theta} \Rightarrow dz =i e^{i\theta}d\theta$$ $$I=i \int_{0}^{2\pi} \log\ |1-e^...
5
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1answer
78 views

Why $P(z)\text{sin}(z)+Q(z)\text{cos}(z)$ has only finitely many non-real zeroes?

Consider entire functions (defined and holomorphic on the whole complex plane) of the form: $$f(z)=P(z)\text{sin}(z)+Q(z)\text{cos}(z),$$ where $P(z)$ and $Q(z)$ are polynomials with real coefficients....
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1answer
20 views

Prove $D(a;R_1,R_2)$ is a connected set

I was solving problems from the start of my Complex Analysis course, and I found this one (the beggining of my course focuses a lot in topology): Prove that $D(a; R_1,R_2)$ is a connected set. The ...
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0answers
35 views

Math topics/subjects that would contribute to a deeper understanding of Engineering concepts

I want to dig deeper into the physical meaning, assumptions and derivations of the mathematical models and formulas I took in mechatronics engineering when I was a student. I don't know where to start,...
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2answers
63 views

Prove that $\oint _{C(0,1)}f(z)dz= \oint _{ D(0,1) }f(z)dz $ [closed]

Define $C(0,1)= \{z: \mid z\mid=1 \}$ $D(0,1)= \{z: \mid z\mid=1, z\neq 1+0i \}$ Prove that, $$\oint _{C(0,1)}f(z)dz= \oint _{ D(0,1) }f(z)dz $$ My try $z=1+0i$ is of measure zero. So, $$\oint _{C(...
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1answer
35 views

Compute $\int_{\gamma} f$

Let $\gamma_1 = S_1 + L - S_2 - L$ and $\gamma_2 = S_1 + L + S_2 - L$, $$S_1(t) = e^{it} , t\in [0,2\pi] $$ $$S_2(t) = 2e^{it} , t\in [0,2\pi] $$ $$ L = [1,2] $$ Let $f(z) = (\cos z)/z$. By writing ...
2
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1answer
28 views

Is $\sqrt{i \cdot \frac{1+z}{1-z}}$ in the Hardy space $H^1(\mathbb{D})$

I am trying to prove that $$\sqrt{i \cdot \frac{1+z}{1-z}}$$ is in the Hardy space $H^1(\mathbb{D})$, where we take the principal branch of the square root. I think that this is true but I am stuck at ...
2
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1answer
44 views

Compute $\int_{\gamma} z\, dz$ for any smooth path $\gamma$ which begins at $z_0$ and ends at $w_0$

Compute the complex line integral$$\int\limits_{\gamma} z\, \mathrm{d}z$$ for any smooth path $\gamma$ which begins at $z_0$ and ends at $w_0$. Confused as to how I am supposed to go about ...
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0answers
16 views

Bound for inverse of gamma function

I am trying to show that $$G(z):= \lim_{n\rightarrow\infty}\frac{n^{-z}}{n!}z(z+1)\cdots(z+n)$$ is the inverse of the Gamma function for $\operatorname{Re}z>0$, and something I need to do is bound $...
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2answers
42 views

Prove f(z) = az + b in the open and connected set G [closed]

Let $G$ be an open connected set in $\Bbb{C}$. Prove that if $f''(z) = 0$ for all $z ∈ G$, then $f(z) = az + b$ for some $a, b ∈ ℂ$.
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0answers
30 views

Proving $|g(z)|$ bounded by $|z|^5$ [duplicate]

If we consider an analytic function g that maps the unit disc $|z|<1$ into itself and has $g(0)=g'(0)=....=g^{(4)}(0)=0$ can we show that $|g(z)| \leq |z|^5$? And if so, when would the inequality ...
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0answers
20 views

Determining complex parameters to lie in the right half-plane

Determine the conditions on the complex parameters $\alpha, \beta$ so that the equation $$z^3 + \alpha z^2 + \beta z +1 = 0$$ has no roots in the right half plane $\Re Z > 0$. My thoughts are to ...
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0answers
20 views

Finding roots on the right half plane

Below is a problem I've been thinking about - it is a textbook problem: Find the many roots the equation $$2z^4-3z^3+3z^2-z+1=0$$ has in the right half-plane $\Re z>0$ I have two thoughts: use ...
2
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1answer
36 views

How to show that the following set $\mathbb{C}/\mathbb{T}$ is compact?

Let $w_1, w_2$ be two complex numbers that are linearly independent over $\mathbb{R}$, i.e., $\frac{w_1}{w_2}$ does not belong to $\mathbb{R}$. Let $\mathbb{T}= \mathbb{Z} w_1+ \mathbb{Z}w_2$. We ...
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0answers
23 views

Proof that the phase of the average of unitary complex numbers is the average phase

I'm quite struggling with the following problem: -Prove that if \begin{equation} re^{i\psi} = \frac{1}{N}\sum_{j=1}^N e^{i\theta_{j}}, \end{equation} then $\psi = \frac{1}{N}\sum_{j=1}^N \theta_{j}$, ...
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1answer
62 views

An onto map to a Unit Circle $|z|=1$ [closed]

An onto map from the critical line $$\frac{1}{2}+it , t\in \mathbb{R}$$ to the unit circle $$|z|=1$$ My try- $$z= \frac{\frac{1}{2}+it}{\frac{1}{2}-it} $$ $$|z|=1$$ But the map is not onto $z=-1$ is ...
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0answers
20 views

Relationship between holomorphy and directional derivatives

Given an entire function $f:\mathbb{C} \to \mathbb{C}$, I would like to prove that the directional derivatives $$ D_\alpha f(z) := \lim_{r\to 0}{\frac{f(z+re^{i\alpha})-f(z)}{re^{i\alpha}}} $$ satisfy ...
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0answers
51 views

Substitution $z=e^{i\theta}$

$$I=\int_{-\pi}^{\pi}ie^{i\theta}f(e^{i\theta})d\theta$$ $$\text{write }\quad e^{i\theta}=z \Rightarrow ie^{i\theta}d\theta=dz$$ $$z=e^{i\theta} ,-\pi\leq \theta \leq \pi$$ So, it becomes $$I= \oint_{|...
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1answer
43 views

Problems with the complex square root

I have a problem understanding the following procedure. ( It's from a script) Consider the domain C[0,$\infty$) and the branch of the logarithm given by $ log(z)=ln(|z|)+i \cdot arg(z)$ ,with $arg(z) \...
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0answers
14 views

Sketch the image under $w=\log(z)$ of the set, $A=\{z\in\mathbb{C}:e^{-\frac{\pi}{4}}\leq\Im(z)\leq e^{\frac{\pi}{4}},\Re(z)=0\}$

Sketch the image under $w=\log(z)$ of the set, $$A=\{z\in\mathbb{C}:e^{-\frac{\pi}{4}}\leq\Im(z)\leq e^{\frac{\pi}{4}},\Re(z)=0\}$$ $u+iv=\log(z)=\ln |z| +iArg(z)$ where $-\pi<Arg(z)\leq\pi$ is the ...
-1
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1answer
60 views

Riemann Xi function $\xi(s)=\xi(0)\prod_{n=1}^\infty (1-\frac{s}{\rho_n})(1-\frac{s}{\overline{\rho_n}}) $ [closed]

Riemann Xi function can be written as $$\xi(s)=\xi(0)\prod_{n=1}^\infty \left(1-\frac{s}{\rho_n}\right)\left(1-\frac{s}{\overline{\rho_n}}\right) $$ Putting $s=1$ $$\xi(1)=\xi(0)\prod_{n=1}^\infty (1-\...
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0answers
17 views

Basics of conformal maps

Currently, I am studying conformal maps i.e. biholomorphic functions. Although I find the theory behind it relatively simple, I do have problems when it comes to showing that a function actually is ...
0
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1answer
20 views

Sketch the image under $w=\log(z)$ of the set, $A=\{z\in\mathbb{C}:e^{-\frac{\pi}{4}}<|z|<e^{\frac{\pi}{4}},\Re(z)>0\}$

Sketch the image under $w=\log(z)$ of the set, $$A=\{z\in\mathbb{C}:e^{-\frac{\pi}{4}}<|z|<e^{\frac{\pi}{4}},\Re(z)>0\}$$ $\log(z)=\ln |z| +iArg(z)$ where $-\pi<Arg(z)\leq\pi$ is the ...
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0answers
29 views

Holomorphic function with given boundary values

I have a complex valued function on the boundary of a set, and want to know if it has a holomorphic extension to the entire set (holomorphic in interior, continuous up to the boundary). In other words:...
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0answers
45 views

Convergence or Divergence of the sum $\sum_{n=1}^{\infty} \frac{2\Re(z_n)-1}{|z_n|^2}, \frac{1}{2}<\Re(z_n)<1 $ [closed]

Convergence or Divergence of the sum $\sum_{n=1}^{\infty} \frac{2\Re(z_n)-1}{|z_n|^2}, \frac{1}{2}<\Re(z_n)<1 $
1
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1answer
43 views

Find all the possible values of $\int_{\gamma} \frac{1}{z^2+1}dz$

a) Let $D = \{z\in\mathbb{C}: z \neq \pm i\}$ and let $\gamma$ be a closed contour in D. Find all the possible values of $\int_{\gamma} \frac{1}{z^2+1}dz$. b) If $\sigma$ is a contour from 0 to 1,, ...
1
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1answer
75 views

Prove that $f(iz)=\overline{f(\bar z)}$ for analytic $f$ satisfies some condition

Let $f$ is entire function in $\mathbb{C}$ satisfying $$ e^{-a|x-y|}\leq |e^{if(x+iy)}|\leq e^{b|x-y|}$$ for some $a,b>0$ . Then to show that $$f(iz)=\overline{f(\bar z)}$$ for all $z\in\mathbb{C}$ ...
2
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0answers
25 views

Questions concerning the proof of Koebe's Kreisnormierungsproblem

I am trying to understand the proof of Koebe's Kreisnormierungsproblem given in Conway's Functions of One Complex Variable II. The theorem states: If $G$ is a non-degenerate finitely connected domain, ...

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