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Questions tagged [complex-analysis]

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, isolated zeros principle...

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1answer
47 views

Why do $x^2+y^2=\left(3+2i\right)^2$ and $x^2+y^2=25$ give the same circle?

Maybe I just been up for too long but I just cant get why: $$x^2+y^2=\left(3+2i\right)^2$$ and $$x^2+y^2=25$$ give the exact same circle, why aren't the graphs different? Shouldn't the second one ...
2
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1answer
22 views

Why $ \Im \frac{1}{1+e^{-s + i a }}=\frac{\sin(a)}{\cos(a)+\cosh(s)} $?

Why this $$ \Im \frac{-2}{1+e^{-s + i a }} $$ equals to this expression: $$\\\ \frac{\sin(a)}{\cos(a)+\cosh(s)} $$ I was trying to evaluate the Fourier transform of a hyperbolic function and my ...
0
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1answer
16 views

Using Contour Integration to solve an integral that holds for all p.

I have been practicing contour integrals and I have come across the following integral and I have been trying to solve it via contour: $$ \int_0^\infty \frac{dx}{x^p+1} = \frac{\pi}{p} \frac{1}{\text{...
0
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1answer
54 views

Complex transformation that transform square into a circle.

Bonjour. I’m looking for a conformal mapping that transform a square into a circle, a cube into a sphere, eventually a rectangle into an ellipse.
0
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0answers
36 views

Does $f=u+iv$ is $\mathbb R-$differentiable $\iff $ $u$ and $v$ differentiable?

I have a theorem that says $f:U\to \mathbb C$ (where $U\subset \mathbb C$ open) is holomorphic on $U$ $\iff$ $f$ is $\mathbb R-$differentiable and satisfy Cauchy-Riemann equation. An other theorem I ...
0
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0answers
13 views

Bounded domains in the complex projective space.

Let $\Omega$ be a bounded convex domain of $\mathbb{C}^3$ and, $$\pi:\mathbb{C}^3\setminus\lbrace0\rbrace\rightarrow \mathbb{C} P^2$$ be the canonical projection into the complex projective plan $\...
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4answers
26 views

Proving $\sum_\limits{n=1}^{\infty}\frac{1}{(n+z)^2}$ converges uniformly

Using the Weierstrass test show that the series $\sum_\limits{n=1}^{\infty}\frac{1}{(n+z)^2}$ converge uniformly on $E=Re(z)\geqslant 1$. This solution was given to me but I am not understanding ...
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0answers
17 views

Proving $\arg(𝑧𝑤)=\arg(𝑧)+\arg(𝑤)$? [duplicate]

algebraic demonstration $\arg(𝑧𝑤)=\arg(𝑧)+\arg(𝑤)$ Proving algebraic demonstration $\arg(𝑧𝑤)=\arg(𝑧)+\arg(𝑤)$
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2answers
20 views

Determining the order of the poles of the function $\frac{1}{\sin z-\sin 2z}$

I encounter a question in my problem sheets, which asks to identify the type of isolated singularities of the following function: $$\frac{1}{\sin z-\sin 2z}$$ Firstly, by trig identities, I can ...
1
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2answers
46 views

Evaluate $\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^z}$ where $\Re(z)>\frac{1}{2}$

I am dealing with the following complex integration: $$\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^z} \quad\text{ where }\quad\Re(z)>\frac{1}{2}$$ But I do not know how exactly to deal with the $z$.
3
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0answers
25 views

Convergence of $\sqrt[n]{(1+x)^n-(1+x)+\sqrt[n]{(1+x)^n-(1+x)+\sqrt[n]{\cdot\cdot\cdot}}}$

If one writes $$1+x=\sqrt{(1+x)^2}=\sqrt{1+2x+x^2}=\sqrt{x+x^2+(1+x)}$$ then one has a recursive definition of the function $1+x$ which can be used to write $1+x$ as the infinite nested radical: $$1+x=...
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1answer
15 views

Behavior of a holomorphic fuction over annulus

I encountered the following problem: Define $ D:=\{ z\in\mathbb C,\ 2<|z|<3 \} $. Let $ f $ be a holomorphic function over $ D $ that is continuous over $ \bar{D} $. (A) Suppose that $ \...
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0answers
22 views

Using the Cauchy-Goursat theorem to prove a statement

For $C$ a simple closed contour in the counterclockwise direction and $C_1$, $C_2$, $C_3$, $C_4$ are subsets in $C$ all in the counterclockwise direction, use the Cauchy-Goursat theorem to prove that: ...
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0answers
14 views

Calculate the Fourier transform using the Airy function [on hold]

Calculate the Fourier transform from (using the Airy function): $$ \Large e^{2ix^3} $$
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0answers
29 views

Entire function which preserves unit disk and fixes $0$ and $1$

Suppose $f \colon \mathbb{C} \to \mathbb{C}$ is an entire function such that $f(0) = 0, f(1) = 1$ and $\vert f(z) \vert \leq 1$ if $\vert z \vert \leq 1$. I want to show that then $f'(1)$ is real and $...
2
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1answer
37 views

How can I minimize the real part of the roots of this function involving both $x$ and $e^x$ terms?

The question I have a function $D(s) = s^2 + c s + k + K_d s e^{-s} + K_p e^{-s}.$ The values of $c$ and $k$ are fixed, but I can choose $K_d$ and $K_p$. How do I choose these two values in order to ...
0
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0answers
16 views

Linear Transformations as conformal mappings

Apologies ahead if this is a duplicate (I'm very certain I've seen a simmilar question but I absolutely cannot find it): Let's consider the scaling (and possible rotation by $\arg(a)$) $$\varphi(z)=...
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1answer
32 views

Expansion of the form $\sum_{n=0}^\infty c_n z^n$ [on hold]

write down an expansion of the form $\sum_{n=0}^\infty c_n z^n$ for $\frac{(a+iz)}{a-iz}$
1
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1answer
34 views

Proving uniform convergence of $f_n(z)=\frac{4n\sqrt{nz}}{3+4n^2z}$

Study the convergence and uniform convergence of the functions $$f_n(z)=\frac{4n\sqrt{nz}}{3+4n^2z},\:z\in[\delta,+\infty],\delta>0$$. $$\lim_{n\to\infty}\frac{4n\sqrt{nz}}{3+4n^2z}=\lim_{n\to\...
1
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1answer
43 views

Given two entire functions such that $\operatorname{Re}(f(z))\le a·\operatorname{Re}(g(z))$, prove that $f(z)=a·g(z)+c$

Let $a\in \mathbb{R}$ and $f, g$ two entire functions such that $\operatorname{Re}(f(z))\le a·\operatorname{Re}(g(z))$. Prove that there exists a constant $c\in \mathbb{C}$ such that $f(z)=a·g(z)+c$ ...
3
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1answer
24 views

Proving $f_n(z)=\frac{nz}{1+n^3z^2}$ converges uniformly

Show that the sequence of functions $f_n(z)=\frac{nz}{1+n^3z^2}$ converges uniformly on the set $E=[1,\infty]$. $\lim_{n\to\infty}\frac{nz}{1+n^3z^2}=0$ so it converges pointwise to 0. So I am going ...
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2answers
53 views

Euler's definition of complex analysis

Can someone help? Need to find the answer of $$\frac{e^{iz} + e^{iz}}{2} \ .$$
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0answers
28 views

Singular points in Complex analysis

Can someone help? Need to find the singular points of $$\frac{z+1}{(z^3)(z^2+1)}$$ where $z$ is a complex variable
1
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1answer
63 views

Show that there is a constant in an analytic function.

Suppose that $f(z)$ is analytic for $|z<1|$ and satisfies $|f(z)|<1, f(0)=0, $and $|f'(0)<1|.$ Let $r<1$. Show that there is a constant $c<1$ sucht that $|f(z)| \leq c|z|$ for $|z|\leq ...
1
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1answer
23 views

Sign convention for Fourier transform and contour integration - example

I was wondering about one (probably trivial) fact during computing the Fourier transform while using contour integral. As an example I have following function: $$f(x)={{1}\over{x^2+a^2}}$$ and its ...
2
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1answer
20 views

Complex integral depending on the chosen path?

Let's say $f:\mathbb C\backslash\{0\}\to\mathbb C$ is holomorphic and $\text{Res}(f,0)=1$. Now if I look for example at the integral $$\oint_{|z|=2}f(z)dz$$ I get confused by the following: The path ...
0
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1answer
76 views

Solve for $x$ when $2^x=-1$

I want to know what is the value of x when: $2^x = -1$ I guess it will be some imaginary number perhaps. I have got the answer as $4.532360129i$. Is that precisely correct or some truncation has ...
0
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0answers
18 views

Tangent of parabola in complex plane [on hold]

How to find equation of tangent of a parabola in complex plane? Supposed the equation of the parabola is |z+1| = Re(z) + c where c is a constant.
2
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0answers
34 views

Fubini Study Geodesics on $\Bbb {CP}^n$

I want to solve the geodesic equation on $\mathbb{CP}^n$ with the Fubini Study metric $$ g_{ij}=\frac{\left(1+{\mid z\mid}^2\right)\delta_{ij}-\bar{z}_iz_j}{\left(1+{\mid z\mid}^2\right)^2} $$ I have ...
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1answer
25 views

Singularity and behaviour at infinity for complex function

I'm suppose to check the singularities and behaviour at infinity. However, I've never seen that and couln't find something about it online. So i have a function $ f(z) = \frac{1}{\exp{(z)} -1)}-\frac{...
0
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1answer
21 views

Laurent series and tensor

Let us begin with the complex vector space \begin{equation} V_{z}=\Big\{\omega\in \mathbb{C}[[z,z^{-1}]]dz\ \mid \operatorname{Res}_{z=0} \omega (z)\Big\} \end{equation} We could define the tensor ...
4
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1answer
67 views

$f : \Bbb{C} \to \Bbb{C}$ is an entire function s.t. $|f(z)| \to \infty$ as $|z| \to \infty$. Prove $f$ is a polynomial.

Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be an entire function such that $|f(z)| \rightarrow \infty$ as $|z| \rightarrow \infty$. Prove that $f$ is a polynomial by following the steps below. (a) ...
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2answers
21 views

Proving $\{f_n(z)=e^{-(z-n)^2}\}$ is uniformly convergent [on hold]

Show that the sequence of functions $\{f_n(z)=e^{-(z-n)^2}\}$ converges uniformly on the set $E=D(0,1)$. $|e^{-(z-n)^2}-0|=|e^{-z^2}.e^{-n^2}.e^{2nz}|\leqslant |e^{-n^2}.e^{2nz}| $ However I do not ...
1
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1answer
33 views

Wirtinger derivatives and Cauchy integral

Suppose a complex valued function $f$ is of class $C^1$ defined on the disk $|z-z_0|<R,$ and let $C_r$ denote the circle $|z-z_0|=r$ with $0<r<R$. Prove that $$\lim_{r \to 0}\frac{1}{2\pi i ...
1
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1answer
25 views

Direct product decomposition of the group of complex roots of unity

I'm studying $p$-adic numbers (Robert's "A course in $p$-adic analysis) and, at page 41, the author states that, for every prime $p$, the group $\mu$ of all complex roots of unity has a direct product ...
5
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1answer
82 views

Is there a solution to this functional equation?

I was going through my old notebooks and I found a sheet of paper with this problem on it. I thought it would be a shame to let such an unreasonably difficult question go to waste, so I decided I ...
0
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0answers
23 views

Branch cut of $\int_0^{\infty} \frac{x^{s-1}}{1+x}$

Why do I need to use a branch cut from $0$ to $\infty$ when evaluating $\int_0^{\infty} \frac{x^{s-1}}{1+x}$? I know there is a pole at $x=-1$ I can't seem to understand the purpose of the branch ...
0
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1answer
27 views

argument of log (z) for different branches

I am studying about the function $\log(z)$ where $\log(z)=\ln r+i\theta$ and as far as I've learnt, defining argument of $\theta$ is base on the given branch. But I am confused how for example for ...
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0answers
17 views

$A\cap\Lambda$ polynomially convex in $\Lambda$

Let $A\Subset\Bbb C^n$ polynomially convex, that is its convex hull $$ \widehat A:=\{z\in\Bbb C^n\;:\;|f(z)|\le\|f\|_{A}\;\;\forall f\in\mathcal O(\Bbb C^n)\} $$ coincides with $A$. Consider then $\...
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0answers
33 views

How to prove that: [on hold]

The linear transformation which maps a triangle into a triangle must be a entire linear transformation.
0
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2answers
63 views

Proving $g'(z)=\frac{1}{2\pi{}i}\int_{C}\frac{g(u)du}{(u-z)^2}$ for $g(z)$ holomorphic in and on contour $C$, and $z$ in $C$'s interior

Prove that if $g(z)$ is holomorphic everywhere inside and on a simple closed contour $C$, taken in a positive sense, and $z$ is any point interior to $C$, then $$g'(z)=\dfrac{1}{2\pi{}i}\int_{C}\...
0
votes
1answer
18 views

Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$?

Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$? where res means the residue of the function? I know $\zeta(s)$ has a pole at $s=1$ but i can't ...
1
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0answers
80 views
+100

Analyticity of this function $\sqrt{\coth^2(a\ z) + \coth^2(b\ z) - c}$

I want to determine the domain of analyticity of this function: $$\sqrt{\coth^2(a\ z) + \coth^2(b\ z) - c}$$ And $$c \in ]0,1]$$ Where $$a,b \in \mathbb{Z} - \mathbb{Z}^+$$ and $a , b$ finite say $a,...
0
votes
1answer
30 views

Analytic function need not have a primitive

I am told that an analytic function need not possess a primitive in its domain of analyticity. However, if $f$ is analytic in the disk $\text{B}(a,r)$, then it has a primitive in that disk. Suppose $f$...
1
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2answers
34 views

Laplace Transform Example: $\mathcal{L}(e^{it}) = \left[ \dfrac{e^{(i - s)t}}{i - s} \right]_0^\infty = \dfrac{1}{s - i}$?

My textbook on Laplace transforms gives the following example: $$\begin{align} \mathcal{L}(e^{it}) &= \int_0^\infty e^{-st} e^{it} \ dt \\ &= \int_0^\infty e^{t(i - s)} \ dt \\ &= \left[...
2
votes
2answers
42 views

Taking Residues of Infinity of Square Roots.

I am looking for worked out exercises of real valued integrals with square roots where ideas of residues at infinity are used. I was hoping that from $$\int_0^1 \sqrt{x} \thinspace dx $$ I could ...
-1
votes
1answer
53 views

Is f(z) = z^n analytic on C, if n is rational

My reasoning why it shouldn't be the case is as follows: Doing a general contour integration of f = z^n along a circle of radius 1 gives us i*int(cos(n+1)t,dt,0,2pi) - int(sin(t+1)t,dt,0,2pi) which ...
1
vote
0answers
25 views

Using the Cauchy Principal Value along a differentiable contour?

I am looking at the following problem from Marsden and Hoffman: Let $ f ( z ) $ be analytic inside and on a simple closed contour $ \gamma $. For $ z_0 $ on $ \gamma $, and $ \gamma $ differentiable ...
0
votes
0answers
24 views

Residue theorem and a two-dimensional integral: not working?

Consider the following integral: \begin{align} \iint_{\mathbb{R}^2}d t\,dT\, \frac{e^{-i(t-T)}e^{-t^2}e^{-{T}^2}}{(t-T-i\epsilon)^2}\,. \end{align} The $i\epsilon$ prescription simply tells me that if ...