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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15 views

How can I prove that $\max_{\partial K}\varphi$ does exist?

Let $\varphi:\Omega\to[-\infty,+\infty[$, where $\Omega\in\Bbb C$ is a domain and $\varphi$ is upper semicontinous, i.e. $\varphi(z_0)\ge\limsup_{z\to z_0}\varphi(z)\;\;\;\forall z_0\in\Omega$. How ...
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13 views

Coupling real functions

I ended up with the following two real functions, that are actually the cosine and sine Fourier transform of other more complicated functions: ...
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15 views

Proof of Levi's problem in Hormander's book

In H$\ddot{\text{o}}$rmander's "An introduction to complex analysis in several variables," he gives a proof of Levi's problem for complex domain by $L^2$ methods. (Theorem 4.2.9) The proof is using ...
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18 views

Roots of of the derivative of a polynomial are in the convex hull of the polynomial

$(def)$: The $\mathbf{Convex}$ $\mathbf{Hull}$ of a set $\{ z_1,\ldots,z_k \} \subset \mathbb{C}$ is the set $$ CH[z_1,\ldots,z_k] = \left\{ z \in \mathbb{C} : z = \sum_{j=1}^k \lambda_j z_j, \; \; 0 ...
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11 views

Let $B = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2| \}$. Show that B is balanced, but that its interior is not.

I have the following definitions. The interior $E^o$ of $E$ is the union of all open sets that are subsets of $E$. A set $B \subset X$ is said to be balanced if $\alpha B \subset B$ for every ...
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1answer
59 views

Show $\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$ if $|z_1| <1$ and $|z_2| < 1$

Show $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$$ if $|z_1| <1$ and $|z_2| < 1$ Consider: $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right|^2$$ ...
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11 views

What is the range of this complex function: $f(z) = 2x^2+(1-x^2)(1+i)$ defined on $|z|\leq1$?

These range problems, I just don't get it. I tried to get this into a form where I could use the fact that $0\leq\theta\leq2\pi$, but I'm just not sure how to get it to that point. Any ${hints}$ ...
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17 views

Does $\frac{1}{z^n}$ have a primitive on $\mathbb{D} - 0$ for $n < 1$?

I'm studying complex analysis from Stein and Shakarachi. There is a question that asks you to evaluate: $\int_{\gamma} z^{n}\ \mathrm{d}z$ for all integer n, where $\gamma$ is any circle centered ...
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3answers
19 views

evaluating a sequence of complex numbers

Let $\{z_n\}$ be sequence of complex numbers such that $$ |z_n - z_m| < \frac{1}{1+ |n-m|} $$ for all $n,m$ Given this information, can we compute $\lim_{n \to \infty} z_n $? Attempt: For sure ...
2
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1answer
22 views

Sketch $ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $

Sketch the following $$ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $$ I have considered this geometrically and ended up thinking that the complex numbers $z$ must satisfy $$0 < ...
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0answers
10 views

Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$

Let $z_1$ and $z_2$ be 2 elements of $C _∞$ . Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$ (the ...
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3answers
80 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
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1answer
48 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
2
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1answer
34 views

Determine the set of points that satisfy the condition $Im \frac{z-z_1}{z-z_2 }=0$ where $z_1$ and $z_2$ are fixed complex number.

Determine the set of points that satisfy the condition $Im \frac{z-z_1}{z-z_2 }=0$ where $z_1$ and $z_2$ are fixed complex number. Since $Im\frac{z-z_1}{z-z_2 }=0$, then there is no imagining part in ...
2
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3answers
32 views

Find taylor of $\psi (z)$ where $(e^z-1)^2=z^2 \psi (z)$ - first 3 terms

I was asked to find the first three terms in the taylor series of $\psi (z)$ around $z=0$ where $(e^z-1)^2=z^2 \psi(z)$ and I'm having a few difficulties. My original idea was to say $\psi ...
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1answer
17 views

Show that a complex map is onto

I consider $\mathbb{C}$ as a real vector space. For $(a,b) \in \mathbb{C}^{2}$, consider the map : $F_{a,b} \, ; \, \mathbb{C} \, \rightarrow \, \mathbb{C}^{\ast}$ such that : $$ \forall z \in ...
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20 views

Continuous complex function from Rudin's Real and complex analysis

Lemma 10.29 from Rudin's Real and Complex Analysis, p. 314 of the third edition states that "if $f \in H(\Omega)$, then $g:\Omega \times \Omega \to \mathbb{C}$ defined by \begin{equation} g(z, w) = ...
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23 views

Understanding Green's Theorem

When looking at Goursat's theorem in complex analysis, I came across the Wiki proof which involves beautiful application of Green's theorem. I saw Greens theorem simply as "connection between line ...
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1answer
42 views

Are complex numbers a trivial lie group of itself? [on hold]

Let $z$ be a complex number, then let's define a map $e^{T(*)}$. Let $w = e^{T(z)}$, where $T$ is some real number. Then is $z$ a lie group of $w$?
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40 views

How to indentify the power series I need to use?

Let $$ f(z) = \frac{1}{(z - 4)(z + 8i)} $$ a) Find the domains where f(z) is valid b) Find its power series at such domains Considering three singularities, I believe the domains are: $$ D_{1} = ...
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1answer
73 views

Find all holomorphic functions $f:\mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}$

Find all holomorphic functions $f:\mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}$ such that $$\limsup_{z\rightarrow 0}\left|\frac{f(z)}{\sin z}\right|<\infty$$ and $$\limsup_{z\rightarrow ...
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65 views

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$ I tried with polar coordinate. So let $z=r(\cos \theta +i \sin \theta)$. the LHS $=|2z|=2|z|=2r$ The RHS, I do some ...
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35 views

Show that $e^z-az^n=0$ has exactly $n$ zeros in $B(0,1)$

Let $a\in\mathbb{C}, |a|>e, n\in\mathbb{N_1}$ I have to show that $e^z-az^n=0$ has exactly $n$ zeros in $B(0,1)$ First, $f(z)=e^z$ and $g(z)=-az^n$ are entire. On $\partial B(0,1)$ we have ...
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1answer
20 views

Mapping of region into a line

Define a function $f:\mathbb{C}\rightarrow\mathbb{C}$, and let $z=x+iy$, then $$f(z)=u(x,y)+iv(x,y)$$ Given that $f$ maps a region into a line, what is the relation between $u(x,y)$ and $v(x,y)$? I'm ...
2
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1answer
22 views

Evaluation of an integral using nonrigorous methods

I was trying to solve the following integral $$ G(\alpha,m,n)=\int_0^{\infty}\cos(2nx)e^{-\alpha x}x^{m-1}dx;n\in N,\alpha>0,m\ge1. $$ By doing a change of variable I brought it to the integral $$ ...
3
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2answers
29 views

Complex Analysis holomorphic function question

I have a Complex Analysis assessment question about holomorphic functions: Let f be a function on a plane and satisfies $f'(z) = f(z)$ and $f(0) = 1$ i) Give an example of a function with this ...
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0answers
17 views

Coupling complex functions

After several calculations I end up with two complex functions: $$g(z)=zA(z)+\overline{z}A(\overline{z})+z^{-1}B(z)+\overline{z^{-1}}B(\bar{z})$$ and ...
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3answers
31 views

Finding $f(i)$ for an Entire Function $f$ that Maps a Line to a Subset of Itself and Sends $1$ to $0$.

Consider the line $L=\lbrace x+iy: x=y\rbrace\subset \mathbb{C}$, suppose that $f:\mathbb{C}\to\mathbb{C}$ is entire and satisfies $f(L)\subset L$. Given $f(1)=0$, find $f(i).$ (Ans: $f(i)=0)$ I am ...
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0answers
10 views

Parametrizing regions of complex plane

Let $\Omega=\mathbb{C}\setminus \lbrace t e^{it} \ \vert t \in \mathbb{R}_{\geq0} \rbrace$ I need to write $\Omega= \coprod_{i=0}^{\infty} R_i$ where each $R_i$ is the region bounded by from $t=2k ...
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1answer
26 views

Complex gamma function

Let $z$ be a complex number with positive real part. By induction on $n$, show that $$ \int_{t=0}^1{t^{z-1}(t-1)^n}dt = \frac{n!}{z(z+1)...(z+n)}. $$ Substitute $t = \frac{u}{n}$ and let $n → ∞$ to ...
4
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1answer
51 views

Find all holomorphic functions $f$ satisfying $f(1-f(z))=f(z)$?

Find all holomorphic functions $f$ (on $\mathbb{C}$) satisfying $f(1-f(z))=f(z)$? First, every constant function $f(z)=w$ is holomorphic and satsfies our condition. Now assume $f$ is not constant. ...
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1answer
21 views

Find all differentiable equations using Cauchy-Riemann equations

Let $z=x+iy$ and $f(z)=u(x,y)+iv(x,y)$. I want to use the Cauchy Riemann equations to find all differentiable functions of the form $$Re( h(z))=2x^2+2x+1-2y^2$$ So I used the C-R equations with ...
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2answers
23 views

Show this is complex differentiable

Prove or disprove $f$ defined on a complex neighborhood of $0$, $$f(x)=\begin{cases} \frac{1}{1+\frac{1}{x}} &\text{ if } x\not= 0\\0 &\text{ if } x=0.\end{cases}$$ is complex differentiable. ...
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1answer
17 views

Convergency of the power series at two points

Consider the power series $$\sum_{n=0}^{\infty}a_{n}(z+3-i)^{n}.$$ The series converges at $5i$ & diverges at $-3i$. Then which is correct ? (a) convergent at $-2+5i$ & divergent at ...
2
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1answer
28 views

Complex Numbers - Finding Limits

$$\lim_{z\to 1+i}\frac{z^4 + 2i}{iz-3}$$ Attempt: I substituted $z = 1+i$ in the numerator and denominator: Since $i^2 = -1$ I got $(1+i)^4 = -4$ So, $$\frac{-4 + 2i}{i-4}$$
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1answer
50 views

All solutions to $ z^{4} = -4 - i16 \sqrt{5} $ [on hold]

I am working on some exercises for my introduction to complex variables class and I have no idea how to solve this question. Given that $ (\sqrt{5} - i)^{4} = z^{4} = -4 - i16 \sqrt{5} $ Find ...
2
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2answers
72 views

Principal value of Fourier Integral

I have tried to find the principal value of $$\int_{-\infty}^\infty {\sin(2x)\over x^3}\,dx.$$ As $ {\sin(2x)\over x^3}$ is an even function, its integral may not be zero in the given limits. I ...
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1answer
23 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega ...
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1answer
21 views

complex analysis differentiation and existence of a point?

If $f(z) = z^3$ prove that there is no point $c$ on line segment $[1,i]$ s.t. $(f(i)-f(1)) / (i-1) = f'(c)$. So differentiating: $$f'(c) = 3c^2$$ $$3c^2 = (f(i)-f(1))/(i-1) = (-i-1)/(i-1) = i$$ ...
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1answer
18 views

Points of a connected open subset joined by a curve

Let $\Omega$ be a connected open subset of $\Bbb C$. Is it necessarily true that any two points of $\Omega$ can be joined by a non-selfintersecting curve, that is, an injective continuous map ...
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2answers
66 views

Determine the region of convergence of series of complex functions

I have this problem. Find the region of convergence of the following series of complex functions $$ \sum_{n=1}^\infty \frac{2^n}{z^{2n}+1} $$ The progress I have made so far is that when n goes to ...
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1answer
22 views

Writing same equation in different forms

I am working with a unit circle with imaginary integration. I know from experience that this can be written as $f(\theta)=\cos t+ i \sin t$ or $e^{i \theta } $ My question would be if i have a circle ...
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1answer
36 views

Why $f(z)=z^2$ is single valued?

Why $f(z)=z^2$ is single valued where $z\in\mathbb{C}$? From definition we have $$z^2=e^{2 \log z}=e^{2(\ln|z|+i(2k\pi+Arg(z)))}$$ I dont get it ;/ Maybe it's getting late.
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1answer
18 views

Proper holomorphic maps and the degree of the map

Suppose f is holomorphic and maps U onto V, both being disks. If f is proper, does this induce a well defined degree for f? And does the converse hold? What are some tools that can help me see if ...
2
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0answers
13 views

Generalization of the Jacobi-Anger expansion to higher harmonics

I know the Jacobi-Anger expansion relation which gives the Fourier development of $e^{i z \cos(\theta)}$ and ${ e^{i z \sin(\theta)} }$, such that $$ \begin{cases} e^{i z \cos(\theta)} = ...
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0answers
13 views

Show that there is no branch of $z^{\frac{1}{n}}$ for $n \gt 1$ in the domain $B(0,1)-\{0\}$

Show that there is no branch of $z^{\frac{1}{n}}$ for $n \gt 1$ in the domain $B(0,1)-\{0\}$ Suppose there is one. Then Since $z^{\frac{1}{n}}=e^{\frac{1}{n}\log z}$, it is equivalent to saying that ...
1
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0answers
9 views

$\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$

Let $f \in M_k(\Gamma)$ not null in $\mathbb{H}$. I want to show that there exists a $C \in \mathbb{C}^{\times}$ and $n\in\mathbb{N}$ with $f = C\Delta^n$. I think one can show that for a $k>0$ ...
1
vote
1answer
23 views

Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$. I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$. I have already shown, that $h$ is ...
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0answers
27 views

Cuscs of a subgroup of $\Gamma$

I'm going to be completely honest about this: I need the solution of this to get permitted to the exam in complex analysis. The topic is not even relevant for the exam and I am absolutely not able to ...
1
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2answers
25 views

Find all singularities of a function and determine its types

Find all singularities of a function and determine its types $$f(z)=\frac{e^{iz}-1}{\sin{z}}e^{\frac{1}{z}}$$ I already showed, that $f$ has poles at points $z=\pi n$ where ...