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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, ...

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4 views

show df(z)/dz=r_hat dot Grad(f )/exp(j*phi). Where z=x+j*y is a complex number and f differentiable at z

show df(z)/dz=r_hat dot Grad(f)/exp(jphi) where z=x+jy is a complex number and f differentiable at z, r_hat is an arbitrary unit vector in x-y plane dot is the dot(inner) product Grad is the gradient ...
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0answers
7 views

Intriguing equality solving Dirichlet problem with Poisson integral

Is it true that $$\frac1{2\pi}\int_{0}^{2\pi}\Re\left(\frac{e^{i\theta}+z}{e^{i\theta}-z}\right)\cos{2\theta}\ d\theta = \frac1{2\pi}\int_{0}^{2\pi}\Re\left(\frac{e^{i\theta}+z}{e^{i\theta}-z}e^{2i\...
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0answers
16 views

Finding range of a complex function

I've been trying to find range of this $f:\mathbb{C}\rightarrow\mathbb{C}$ function: $$f(z) = z+2\overline{z}+z\cdot\overline{z}+iz$$ I tried writing it out as: $$c+di =a+bi-2a-2bi+a^2+b^2+ai-b$$ ...
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3answers
33 views

$\int^0_{-\pi} {i e^{2it}\sin(2t)}$ by parts?

By integrating by parts the following: $$\int^0_{-\pi} {i e^{2it}\sin(2t)}dt$$ with $$u=\sin(2t); v'=ie^{2it}$$ $$u'=2\cos(2t);v=\frac{1}{2}e^{2it}$$ I get: $$\int^0_{-\pi} {ie^{2it}\sin(2t)}dt = ...
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2answers
17 views

Complex Anti-derivative of tan(z)

Show $tan(z)$ has a complex anti-derivative on $S=\mathbb{C}\backslash((-\infty,-\pi/2]\cup[\pi/2,\infty))$ If F(z) is the complex antiderivative of $tan(z)$ on $S$, find $F(i)$, if $F(0)=0$ I ...
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1answer
18 views

Calculate complex curve integral along rectangle

Determine the line/contour integral of: $$\int_{\gamma}\frac {z}{z^3+1}dz$$ where $\gamma$ is the boundary of a rectangle defined for $0\leq x\leq 2$ and $-2\leq y\leq 2$. I am almost certain we ...
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0answers
18 views

Is $\sum^\infty_{n=1}\mu(n)z^n$ a lacunary function?

Let $\mu(n)$ be the mobius function. Then, is $$f(z)=\sum^\infty_{n=1}\mu(n)z^n$$ a lacunary function? Clearly, the series converges in the open unit disk. Since I have read from somewhere (maybe ...
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2answers
21 views

If $f$ has an isolated singularity at $z_0$ and if $\lim_{z \to z_0}(z-z_0)f(z) = 0$, then the singularity is removable.

If $f$ has an isolated singularity at $z_0$ and if $\lim_{z \to z_0}(z-z_0)f(z) = 0$, then the singularity is removable. This is the Riemann Principle of removable singularity and I am using the ...
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19 views

Laplace inverse of $\cos(s)$ [on hold]

I need the solution of this question What is the Laplace inverse of $\cos(s)$
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0answers
39 views

Prove $f(z) := b_0 + b_1z + b_2z^2 + …$ defines … an analytic function satisfying the equation $f(z) = 1+zf(z) + z^2f(z)$.

My work so far is as follows: $1 + zf(z) + z^2f(z) = 1+z(b_0+b_1z+b_2z^2+...)+z^2(b_0+b_1z+b_2z^2+...)$ Rearranging, we get this equals $1+b_0z+(b_0+b_1)z^2+(b_1+b_2)z^3+...$, which is of the form $...
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1answer
22 views

Can there exist a second cube root of $z$ located in the first quadrant?

Suppose $w$ is located in the first quadrant and is a cube root of a complex number $z$. Can there exist a second cube root of $z$ located in the first quadrant? I'm not sure of this question. I'm a ...
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1answer
21 views

Suppose $n$ denotes a non negative integer. Determine the values of $n$ such that $z^n=1$ possesses only real solutions.

Suppose $n$ denotes a non negative integer. Determine the values of $n$ such that $z^n=1$ possesses only real solutions. My attempt: Let $w\in \mathbb{C}$ such that $w=\cos0+i\sin0$. and $z=p(\cos\...
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1answer
44 views

Let $f$ be the disk $D[0, 1]$ be a holomorphic function. Show the following

I was able to prove a) using the limit of $g$. I know that b) has something to do with Cauchy's formula, and I tried to use the RHS to get the LHS, but got stuck after expanding it. I also tried ...
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2answers
27 views

Determine the number of point $z \in \mathbb{C}$ such that $(2z+i\overline{z})^3=27i$

I've just learned complex numbers in Mathematical Analysis 1, and I'm stuck in the following problem: I would like to determine the number of point $z \in \mathbb{C}$ such that $(2z+i\overline{z})^3=...
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0answers
24 views

Region of convergence: Complex vs Real valued power series.

The real valued power series: $$\sum_{n=0}^\infty a_n(x-x_0)^n$$ converges when: $|x-x_0|<R$ and diverges when $|x-x_0|>R$. I want to find the region of convergence for the complex valued ...
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0answers
13 views

limit and continuity on complex plane [on hold]

let $f(z)$ be defined by f(z)=["]z if $z\neq0$, $f(z)=1$ if $z=0$. At which point $f(z)$ have a limit, and at which point it is discontinuous? Which of discontinuous of $f(z)$ are removable?
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1answer
22 views

Complex anti-derivative and the path integral property

I am confused how the existence of a complex anti derivative on an open set implies the path integral property. The proof of this is shown here and it is very simple. Is this not a trivial counter ...
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0answers
11 views

showing existence of a sequance of a compact sets [duplicate]

There is $\Omega \subseteq \Bbb C$ an open set, I need to show there exist sequance ${K_n}_{n=1}^{\infty}$ of a compact sets, such that: 1) for every $n\in \Bbb N$ ,$K_n$ is contained in the interior ...
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1answer
29 views

Is there a holomorphic bijection from the left-hand side of the complex plane to the unit disk with this property?

I am trying to find a complex mapping from the left-hand side of the plane to the unit disk, such that $$\lvert z_1\rvert\gt\lvert z_2\rvert\iff \lvert f(z_1)\rvert\gt\lvert f(z_2)\rvert$$ I wasn't ...
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0answers
19 views

Proving that a complex power series is differentiable

Let $p(z)=a_0+\dots+a_nz^n, n\geqslant 1, a_n\neq 0.$ I want to show that $f:\mathbb{R}\rightarrow\mathbb{R}$ is a complex differentiable function, where $f=1/p(z)$. I already showed that there is an $...
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1answer
34 views

Start for a proof of the fundamental theorem of algebra [duplicate]

Let $p(z)=a_o+\dots+a_nz^n$, with $n\geqslant 1$ and $a_n\neq0$. Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be defined by $f(z)=1/p(z)$. I need to show that there is an $R>0$ such that \begin{...
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1answer
33 views

Finding an argument of sum of two complex hyperbolic functions

I want to find the absolute value and the argument of the following complex number $$ -\cosh[\sqrt{z_1}]+\frac{z_2}{\sqrt{z_1}}\sinh[\sqrt{z_1}] $$ where $z_1$ and $z_2$ are two complex numbers. Can ...
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0answers
11 views

Fitting a conformal / holomorphic function

Suppose we have some 2D points $x_i$ (which we may take to be complex numbers) and some corresponding 2D points $y_i$. We seek a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x_i)\approx ...
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0answers
34 views

Show that the function has a derivative only at points that lie on the x-axis. [on hold]

Show that the function $f(z)=x^2+y^2+i2xy$ has a derivative only at points that lie on the x-axis.
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30 views

Use the Cauchy-Riemann conditions, to show that the following functions are differentiable or not. Why? [on hold]

Use the Cauchy-Riemann conditions, to show that the following functions are differentiable or not. \begin{align} (1) \quad &f(z)=-2(xy+x)+ \mathrm i(x^2-2y-y^2). \\ (2) \quad &g(z)=e^{-\...
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1answer
27 views

Poles of $f(z)=\frac{\sin(\frac{\pi z}{2})}{\sin(\pi z)}$

Consider the complex valued function $$f(z)=\frac{\sin(\frac{\pi z}{2})}{\sin(\pi z)}$$ I am trying to investigate the poles at integers points. That is points where $z=n,n\in \mathbb{Z}$ Case 1: ...
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0answers
18 views

The cross ratio $ (z_1,z_2,z_3,z_4)$ is real iff the four points lie on a circle or a straight line

It's written in Alfors Complex Analysis that, for a proof of the above, "we need only show that the image of the real axis under any linear transformation us either a circle or a straight line. Indeed,...
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1answer
24 views

Reflection about a line as a möbius transformation

I am trying to find a matrix representation in Mat$_{2×2}(\mathbb C)$ for a reflection about a line $z=z(t) = a+bt$ where only $t$ is restricted to be in $\mathbb R$ as a parameter. I am thinking ...
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1answer
13 views

Let $D=\{z\in\mathbb{C} : |z+i|+|z-i|=1\}$ find the set $w.D=\{wz:z\in D\}$ where $w=i$.

Let $D=\{z\in\mathbb{C} : |z+i|+|z-i|=1\}$ find the set $w.D=\{wz:z\in D\}$ where $w=i$. My attempt I go to proceed to describe the set $D$. Let $z=x+iy$ then $\sqrt{x^2+(y-1)^2}+\sqrt{x^2+(y+1)^...
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0answers
52 views

What is a common framework for these divergent sums?

If you expand $2^x$ using a finite difference series you end up with the formula $$ 1 + x + \frac{1}{2!}x(x-1) + \frac{1}{3!}x(x-1)(x-2) ... = \sum_{n=0}^{\infty} \frac{(x)_n}{n!} $$ Now these ...
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1answer
22 views

Complex logarithm multiplication/exponent rule

I am trying to show that $\log(i^2) \ne 2\log(i)$ and that $\log(i^{1/2}) = 1/2\log(i)$ but I am having difficulties. I know that $\log(z) = \ln|z| + i\arg(z)$ but when playing around with these ...
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0answers
32 views

$z=0$ is not a local maximum of $|p(z)|$

How do I show that $z=0$ is not a local maximum of $|p(z)|$ where $p(z)=a_0+a_1z+\cdots+a_nz^n$ if $a_i \neq 0$ for some $i>0$. My try: We want to find $z$ in some $\epsilon$ neighbourhood such ...
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1answer
9 views

Uniform limit of complex difference quotient

Let $\mathcal{C}_0([0,1])$ be the set of continuous complex-valued functions defined on $[0,1]$ such that they vanish at the origin and let $\|\cdot\|_{MAX}$ denote the norm $$\|x\|_{MAX} = \max_{t\...
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1answer
23 views

Difference real and complex fourier series

I'm working on fourier series and I'm trying to compute the fourier transformation for the $2\pi$-periodic function of $f(x)=x^2$ with $x \in [-\pi,\pi]$. Now with the real way, that is $$f(x) \sim \...
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0answers
30 views

Solving functional equation $\sin f = f^2-3if+\pi$

I want to find all (entire) solutions to the equation $$\sin f = f^2-3if+\pi.$$ Using the identity theorem, I was able to show that if a solution exists, it must be constant. Therefore, all that is ...
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1answer
35 views

Show that $e^z$ is continuous on $\mathbb{C}$

I know that $e^z$ is continuous on $\mathbb{R}$, but how would I show this rigorously on $\mathbb{C}$ using the $\epsilon - \delta$ definition of continuity? I know how to begin: If $|z - z_0| < \...
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0answers
20 views

Extending function analytically

I'm trying to understand why the following is true: Suppose you have a complex holomorphic function such that $f(0)=0$, $0<|f'(0)|=\lambda<1$. Then there exists $r>0$ such that the sequence $\...
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1answer
19 views

Is this function continuous on $\mathbb{C}$?

Let $z \in \mathbb{C}$. On what subsets is this function (dis)continuous? $g(z) = \frac{sin|z|}{z}$ My attempt: Write $z = x + iy$. Then: $g(z) = \frac{\sin|z|}{z} = \frac{sin(x^2 + y^2)}{x+...
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2answers
83 views

Find the zeros of $f(z)=z^3-\sin^3z$

I want to find the zeros of $f(z)$, $$f(z)=z^3-\sin^3z$$ My attempt $f(z)=0$ $z^3-(z-z^3/3!+z^5/5!-\dots)^3=0$ $z^3-z^3(1-z^2/3!+z^4/5!-\dots)^3=0$ $z^3[1-(1-z^2/3!+z^4/5!-\dots)^3]=0$ So $z=0$ ...
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1answer
37 views

How to solve a two variable equation

Suppose $m,n$ are positive constants,there is an equation $m|\lambda_1|^2+|\lambda_2|^2-n\lambda_1\bar{\lambda_2}-n\lambda_2\bar{\lambda_1}=0$,where $\lambda_1,\lambda_2$ are nonzero complex numbers. ...
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3answers
20 views

Existence of a holomorphic function $f$ on the unit disc, such that $f(1/n) =\frac{(-1)^{n}}{n^2}$ for any integer $n>1$

Does there exist a holomorphic function $f$ on the unit disc, such that $$f\left(\frac{1}{n}\right)=\frac{(-1)^{n}}{n^2}$$ for any integer $n>1$? Now, a function like $$g(z) = z^2(-1)^{1/z}$$ ...
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0answers
17 views

Residue and Laurent series for an integral involving exponential and square root.

In another post, I am trying to compute this integral: $$\int_{-1}^1 \frac{e^{bx}}{(x-a)^2}\sqrt{1-x^2}dx\quad b>0,\;a>1$$ I try to tackle it using the residue theorem. When I write it with ...
2
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0answers
45 views

About a proof on Hubbard's Teichmüller Theory

I'm reading the book "Teichmüller Theory and Applications to Geometry, Topology and Dynamics" by John Hubbard and I have the following question. The proposition 7.4.4 says: Let $\varphi,\psi$ be ...
2
votes
1answer
44 views

Show that $f_a(z)=z+a-e^z$ has only 1 zero in $Re(z)<0$ and this zero is $<0$. $(a>1)$

I am trying to use Rouche's Theorem somehow but I can't seem to be able to find a proper function to compare $f_a(z)$ with. I tried $g(z)=z+a$ but then I can't deal with the $e^z$ term. Any ...
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0answers
26 views

Extending Lacunary Series beyond their disks

I've been for the past year and half fascinated by the lacunary series $f(z) = \sum_{n=0}^{\infty} z^{2^n}$. This function obeys the following equation inside the unit disk. $$f(z^2) = f(z)-z$$ And ...
1
vote
0answers
42 views

Equalities using Euler's formula

I have $$ e^{1+2 \pi i}= e^{1} e^{2 \pi i} = e,$$ then $$e=e^{1+2 \pi i} = ( e^{1+ 2 \pi i})^{1+2 \pi i} = e^{(1+2 \pi i) (1+2 \pi i)}= e^{1+4 \pi i - 4 \pi^2}= ee^{4 \pi i}e^{-4 \pi^2}= ee^{-4 \...
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1answer
19 views

Conformal mapping interpretation

It says on Alfors, Complex Analysis page 73, chapter 2.3 that: Suppose that an arc $\gamma$ with the equation $z = z(t), \alpha \leq t \leq \beta$, is contained ina region $\omega$, and let $f(z)$ be ...
1
vote
1answer
19 views

Complex Analysis - How a region's boundary changes under a complex function

Hi, I'm trying to solve this question. I tried plugging in the outer boundary for G (left picture), which is just the unit circle in the complex plane, but I can't simplify it into the form I want $\...
3
votes
2answers
23 views

If a set contains all accumulation points then it is closed

It is a question from my complex analysis courses; If a set contains all accumulation points then it is closed Our accumulation point definition is “if a point is an accumulation point of set $S$, ...
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0answers
28 views

Find the argument

Find the argument $$\begin{align} &A. \left(\frac{\sqrt{3}}2+\frac i2 \right)^7 \\ &B. (11 − i 11\sqrt{3})^9 \end{align}$$ I get $\frac{7\pi}6 +2kπ$ for A. I just want to make sure that ...