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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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
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1answer
36 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
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1answer
37 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 ...
4
votes
2answers
33 views

Flipping sign of $i$s

Why do we flip the signs of all $i$ s in a complex number when we want to take the conjugate of it? I mean, conjugating means making $x + iy$ into $x - iy$, but given a number of the form: $$\frac ...
3
votes
1answer
85 views

Find all holomorphic diffeomorphisms $f:\mathbb{CP}^1\to\mathbb{CP}^1$

The complex projective line $\mathbb{CP}^1$ is the complex manifold defined by the quotient of $\mathbb{C}^2-\{(0,0)\}$ by the relation $z\sim w$ if $z=\lambda w$ for $\lambda\in\mathbb{C}-\{0\}$. I ...
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4answers
35 views

Proving arg(z/w)=arg(z)-arg(w)

I need to prove that $$arg\left(\frac{z}{w}\right)=arg(z)-arg(w)$$ However, I am a little stuck as to how to go about this. I know the proof for $arg(zw)=arg(z)+arg(w)$ happens by letting ...
3
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0answers
51 views

Complex Analysis (Complex Mapping) stuck on professor's method of simplification in math notes

I'm having an exam this university semester and need some help with my math notes. I've come accross some problems with the section "Complex Mapping." Link to Image of my Notes: Click Me (see first ...
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0answers
27 views

Ordering in the complex plane [closed]

Explain why there is no ordering ≺ on the complex numbers that satisfies the usual properties of the relation < on the real numbers I just need help starting this problem. I'm not sure what ...
3
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0answers
42 views

Finding Laurent Series of a function

I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
2
votes
0answers
21 views

Claim about the z-transform of a discrete function

Claim: $\lim_{k\to\infty} x[k]$ exist and if finite is $X(z)$ the Z-transform of $x[k]$ has no pole in $|z|>1$ and at most 1 pole at $z = 1$ Attempt: \begin{align*} X(z) &= ...
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1answer
17 views

Harmonics conditions for a plucked string

Given a plucked string which is taken on the interval $[0,\pi]$, and it satisfies the wave equation with $c=1$. The initial position of the string is: $\ f(x) = \frac{xh}{p}$ ($0\leq x\leq p$), and $\ ...
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0answers
133 views

How many complex functions reduce to a given x-y function?

A 2D or x-y coordinate function has a complex analog, which is formed by replacing x with with the complex variable z. That function can then be separated into real and imaginary parts. Graphing the ...
6
votes
1answer
128 views

Can $f(g(x))$ be a polynomial?

Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions. Is it possible that $f(g(x))$ is equal to a polynomial ? edit Some comments : I was thinking about iterations. So for instance ...
2
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1answer
14 views

Determining uniform convergence of complex power series

I'm working on some practice problems for my complex analysis course, and I'm having trouble with uniform convergence. The question asks whether the following series converges uniformly for ...
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0answers
34 views

problem in complex analysis about the modulus [closed]

Let $z$ and $w$ be any two complex numbers such that $|z|\leq 1, |w| \leq 1$ and $\bar w z$ $\not= 1$. Prove that $$\frac{|(w-z)|}{|(1-\bar w z)|} \leq 1,$$ with equality if and only if $|z| = 1$ or ...
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0answers
28 views

Does uniform convergence on $D\subseteq \mathbb{C}$ imply uniform convergence on all subsets of $D$?

Let $f_n:D\rightarrow \mathbb{C}~\forall n\in \mathbb{N}$. If $(f_n)_{n\in\mathbb{N}}$ converges uniformly on $D\subseteq \mathbb{C}$ against $f:D\rightarrow \mathbb{C}$, does $(f_n)_{n\in\mathbb{N}}$ ...
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1answer
62 views

Find roots of $ω^x+(ω^x)^2+1=x$ [closed]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
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0answers
14 views

Find the equation of an ellipse in a complex plane.

Find an equation for the ellipse that passes through $3+7i$ if its foci at $i$ and $-1$. This is what i have so far. I know that the equation of an ellipse is $|z-p| + |z-q| = c$. so $|z-p| + |z-q| = ...
2
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3answers
74 views

Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
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0answers
41 views

Solve equation $ω^x+ω^{2x}+1=x$ [closed]

We have that to solve $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right.
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1answer
30 views

Complex Analysis (Limits)

Let $a, b$ be complex numbers. Use the definition of a limit directly (not just the properties of limits) to prove that $$ \lim_{z \to z_0}az + b = az_0 + b. $$ Sorry for the wrong notation, I do ...
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1answer
45 views

Is a connected Reinhardt Domain which containg $0$ necessarely a polydisc?

I'm studying several complex variables basics. Roughly speaking: call $D\subseteq\Bbb C^n$ the set of points in which a given power series $$ \sum_{\alpha\in\Bbb N^n}a_{\alpha}(z-z_0)^{\alpha} $$ ...
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1answer
19 views

Schwarz Reflection Principle vs. Analytic Continuation

Analytic continuations are unique on simply connected domains: $$F,F':\Omega\to\mathbb{C}:\quad F\restriction=F'\restriction\implies F=F'$$ Schwarz reflection principle offers analytic continuations ...
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2answers
31 views

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: ...
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2answers
38 views

complex numbers equality question

Let $a$ be given a complex number. Show that $$\left|\frac{z-a}{1-a^*z}\right|=1$$ for $z$ with $|z|=1$ and $a^*z\neq 1$. If $|z|=1$, that means $z$ can be equal to $i$, $-i$, $1$ or $-1$ right? ...
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1answer
65 views

Mean Value Theorem for $f: \mathbb{R} \rightarrow \mathbb{C}$

Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a continuous and differentiable function on $[a, b]$. Then does there exists a $c \in (a,b)$ such that $$\frac{|f(b) -f(a)|}{b - a} \leq |f'(c)|?$$
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1answer
18 views

Express $\sin(z)$ and $\cos(z)$ in Rectangular Form

"Express $\sin(z)$ and $\cos(z)$ in rectangular form." For $z \in \mathbb{C}$ (complex numbers), we have defined \begin{equation} \sin (z)=\frac{e^{iz}-e^{-iz}}{2i} \end{equation} and ...
0
votes
2answers
35 views

Plotting on a complex plane

I'm very confused how you would plot the relationship $|z-4| \leq |z|$. I tried to change it in form which could become $-|z|\leq|z-4|\leq|z|$ and I guess the same can be done for z-4. But I don't ...
0
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1answer
16 views

Computing Principal Logarithm on Different Intervals

Compute the principal logarithm of a complex number $z=\sqrt{3}+i$ using $\mathrm{Arg}(z) \in [0,2\pi)$ and $\mathrm{Arg}(z) \in (-\pi,\pi]$. Wikipedia shows how the answer can be different for the ...
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1answer
9 views

Question about determining accumulation points

So far the way I have determined accumulation points of given sequences or relations has been by drawing them out. However I would like some clarification to see if my thinking is correct or not. a) ...
0
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2answers
16 views

Defining domain in complex plane

I am asked to define the domain for the following given that $z=x+iy$: $a) \quad f(z) = \dfrac 1 {z^2 + 1}$ $b) \quad f(z) = \dfrac 1 {1 - |z|^2}$ How would this be different from a normal domain ...
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1answer
24 views

Problem with complex derivative

I have to find all points where function $f(z)=\mathbb{Re}z \cdot |z|$ is complex differentiable. CR equations arent satisfied in points $\mathbb{C} \setminus \{0\}$. So Im calculating derivative at ...
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1answer
76 views

Prove $\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$

I have to prove that for $f(x,y)=e^{-xy^2}\sin(x)$ and $\forall b>0$ we have $$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$$ I've tried to ...
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0answers
16 views

Enstrom and kakeya theorem

i need some numerical application of enestrom Kakeya thereom ? in fench je cherche une application du theoreme d enstrom et kakeya application numerique ou une application dans un domaine de ...
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1answer
47 views

Taylor expansion of $1/(1+z)$

How do I obtain the Taylor expansion of $$\frac{1}{1+z}$$ about $a=i$ please? Do I just expand the series using the binomial expansion?
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1answer
57 views

Does every group have a lie algebra? [closed]

Does the complex number group {$C, *$} have a lie algebra? When does a group have a lie algebra?
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2answers
44 views

Trivial complex analysis proof

Question: Prove that if $z, w, v ∈ C$ and $zwv = 0$ then at least one of $z$, $w$ and $v$ must be $0$. My thought was that first, I would assume that $zwv=0$ and that $z,w,v\neq0$ This leads to a ...
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1answer
43 views

Conditions To Make Complex Numbers $z_1, z_2, z_3, z_4$ Vertices of a Square

Let $z_1,z_2,z_3,z_4\in\mathbb C$ be distinct. State conditions in terms of computation of complex numbers, which make $z_1,z_2,z_3,z_4$ vertices of a square (in the counterclockwise direction). ...
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1answer
46 views

Limit involving branch point of a complex function

I am having trouble with the following problem : If we restrict ourselves to that branch of $f(z)= \sqrt{z^2+3}$ for which $f(0)=\sqrt 3$ , prove that $$\lim_ {z\to 1}\frac{\sqrt{z^2+3}-2}{z-1} = ...
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0answers
31 views

Prove there is no branch of arg $z$ on $0 < z < 1$.

If $G$ is an open connected subset of $\mathbb{C}$ that does not contain the origin, we call a continuous function $\alpha$ satisfying $\alpha(z) = \text{arg} z$ for all $z \in G$ a branch of arg $z$. ...
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1answer
31 views

What does “$C^{\infty}$” convergence mean?

I'm studying first notions about several complex variables. As a consequence of the (generalized form) of the Cauchy esteem for holomorphic functions, the book says that in the space $\mathcal ...
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1answer
32 views

Composition of harmonic and holomorphic function

Simmiliar to this question my problem is as following: If $u$ is harmonic, and $f$ is holomorphic function, are $u \circ f$ and $f \circ u$ harmonic? I tried to do it like this: $$\Delta (u \circ f)= ...
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2answers
24 views

Isolated singularity is removable iff $\lim\limits_{z\to z_0} (z-z_0)f(z)=0$

Could someone explain a step in the following proof? Theorem An isolated singularity $z_0$ of $f$ is removable if and only if $\lim\limits_{z\to z_0} (z-z_0)f(z)=0$. Proof ($\Leftarrow$) ...
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2answers
70 views

Proof of $\cos(y)$ and $\sin(y)$ using $e^{iy}$

I need to use that $e^{iy} = \cos y + i \sin y$ (for $y \in \mathbb{R}$) to prove that $$\cos y = \frac{e^{iy}+e^{-iy}}{2}$$ and $$\sin y = \frac{e^{iy}-e^{-iy}}{2i}$$ I'm really clueless, any ...
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1answer
27 views

Holomorphic function is zero on an analytic set then $df=0$.

Assume we have an homomorphic function $f:U\rightarrow \mathbb{C} $ which is holomorphic on the open set $U$ of $\mathbb{C}^n$. Assume there is $V\subset U$ analytic and that $f$ restricted to $V$ ...
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1answer
20 views

Green's function for Dirichlet problem on a half disk

Let $D=\{z=(x,y):x^2+y^2<R^2, y>0\}$ be the half disk with radius R. Then if we consider the Dirichlet problem on this domain, i.e., we want to find $$ \Delta u=0, ~~z\in D,\\ ...
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1answer
42 views

Check if $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ is compact

I want to check, if this set is compact: $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ Thoughts: $z:= a +bi$ real part $a$ is ...
0
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1answer
19 views

Deriving definition of the complex logarithm

Given that: $$z = Re^{i\theta} = R(a + bi) = R\left( \cos(\theta) + i\sin(\theta) \right)$$ In its polar form. $$\log(z) = \log(R) + i\theta$$ $$|z| = \sqrt{(Ra)^2 + (Rb)^2} = R\sqrt{a^2 + b^2} ...
2
votes
0answers
28 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
2
votes
2answers
49 views

Exercise: Evaluating integration $\int_{|z|=r} \frac{1}{(z-a)(z-b)}dz$, $|a|<r<|b|$

This is an exercise from Stein-Shakarchi's Complex Analysis: evaluate integration $$\int_{|z|=r} \frac{1}{(z-a)(z-b)}dz, \,\,\,\, |a|<r<|b|. $$ The problem I am facing is the following. It is ...