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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7 views

changing limit with integral sign

I have the following question: For $z$ not in the real interval $[0,1]$, let $f(z)=\int_{0}^{1} \frac{t^{2}dt}{t-z}$. Show that $f$ is differentiable. I got $\lim_{z\rightarrow w}\,\,\int_{0}^{1} ...
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1answer
25 views

In choosing a keyhole contour to compute a complex valued integral, is the inner loop positively oriented?

If the outer circle starts in a counterclockwise direction, i.e., positively oriented, and turns into the straight line, which turns into a small circle (a loop, technically, since it doesn't fully ...
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1answer
31 views

Calculating $f'(z)$ for a complex function $f$.

So im solving some exercices in which they asi to find where $f$ is analytic and then finding $f'(z)$. So I know that if a function doesnt satisfy Cauchy Riemman then it is not analytic. I also know ...
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15 views

Gram-Schmidt procedure on complex waveforms

I am trying to derive an orthonormal basis for the following complex wave forms using Gram-Schmidt procedure $$s_1(t)=\sqrt{2E\over T}e^{j\phi{[n-1]}} e^{j\pi t\over 2T},0\leq t\le T$$ ...
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0answers
25 views

identity theorem with a triangular region

I have the following question: Let D be a triangular region in the complex plane with boundary $B$. Suppose that $f$ is analytic in $D$ and continuous on $D\cup B$. If $f$ is constant in one side of ...
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1answer
52 views

Books on complex analysis (Ahlfors, Conway and Lang)

To make my question slightly different from others, I would like to know how would you rate on the complex analysis books by Ahlfors, Conway and Lang? I had a basic course on complex analysis during ...
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3answers
55 views

Show that a function from a Riemann Surface $g:Y\to\mathbb{C}$ is holomorphic iff its composition with a proper holomorphic map is holomorphic.

I'm trying to show the following: Let $f:X\to Y$ be a proper holomorphic map between connected, non-empty Riemann Surfaces. Show that a map $g:Y\to\mathbb{C}$ is holomorphic if and only if its ...
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2answers
35 views

Finding where $ f$ is analytic

Find all the $ z \in \Bbb {C}$ where $f (z)=\frac {z+1}{z-1}$ is analytic. So im getting started with complex analysis, and being used to doing some real analysis im getting messed up. So in the ...
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12 views

An analytic function on the open unit disk with a zero of order k

I have the following question: Let $f$ be an analytic function from the open unit disk to itself. Assume that $f$ has a zero of order $k$ at zero where $k\geq 1$. Prove that $|f(z)|\leq |z|^{k}$ for ...
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86 views

Is this elementary proof of fundamental theorem of algebra correct?

There is an elementary proof which considers the polynomial $ p(z) $, $z \in \mathbb{C}$ as a function of $ (r,\theta) $ where $ z= r e^{-i \theta} $, $r\ge 0$ . There are two assumptions- Assumption ...
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1answer
20 views

Improper integral parametrised in complex variable: when is it holomorphic?

Suppose we are considering the following integral: $$ I(s) = \int_1^\infty t^{s-1}e^{-t\lambda}\;dt $$ where $s \in \mathbb{C}$ and $\lambda > 0$ is a fixed constant. I would like to know when this ...
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1answer
25 views

Isolated singularities: removable vs poles

I understand what the singularities are, but I am having trouble establishing them in what I feel is a formal fashion. Take these two questions I am working on. $$\frac{z^4 - 2z^2 + 1}{(z-2)^2} $$ ...
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2answers
48 views

Contour integration of exponential function [on hold]

How to solve this integral with residues method? $$\int_0^\infty \frac{e^{ixp}}{x^2+1+i}dx$$
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1answer
15 views

Unform convergence on a bounded subset of the complex plane

What does it mean when someone says that the exponential series converges uniformly on every bounded subset of the complex plane. I know the definition of convergence, uniform convergence of sequence ...
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1answer
20 views

if $\operatorname{Res}_{z_0}f = 0$, then $f$ has a primitive in some deleted neighborhood of $z_0$

Let $z_0$ be an isolated singularity of $f$. Prove that if $\operatorname{Res}_{z_0}f = 0$, then $f$ has a primitive in some deleted neighborhood of $z_0$ I know that if we assume $f$ has a ...
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0answers
25 views

Is this proof correct? Show that the residues are equal

I think I solved the following question, but was unsure about a couple of details. Let $f$ be meromorphic in a neighborhood of $0$, $\phi$ analytic in a neighborhood of $0$, such that $\phi(0) = ...
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0answers
44 views

Behaviour of $\zeta(s)$ near $s=1$

I would appreciate if somebody could run this over and see if it works out? any suggestions or pointers would be appreciated. I denote the standard eta function $\eta$ by $\zeta^{*}$. I have not used ...
6
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0answers
30 views

Uniform limit of injective analytic functions is injective

I'm stuck on the following problem: Let $f_n$ be a sequence of injective analytic functions on the unit disc $D$ such that $f_n$ converges uniformly to $f$ on compact subsets of $D$. Show that ...
3
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1answer
41 views

$f: \Omega \rightarrow \Omega$ bounded, $f(z_0) = z_0$, show $|f'(z_0)| \leq 1$

I'm stuck on the following problem: Let $\Omega$ be a bounded domain, and $f: \Omega \rightarrow \Omega$ analytic such that $f(z_0) = z_0$. Show that $|f'(z_0)| \leq 1$. What I did so far was ...
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1answer
29 views

Why does Liouville's Theorem imply that there are no 1-1 holomorphic maps from $\mathbb{C}$ to $D(0,1)$?

Liouville's Theorem say that every bounded entire function is constant. I can't see why one of the consequences of it is there is no 1-1 holomorphic maps from $\mathbb{C}$ onto $D(0,1)$, where ...
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1answer
44 views

In which sense does Cauchy-Riemann equations link complex- and real analysis?

On page 12 of Stein, Shakarchi textbook 'Complex analysis', the authors state that the Cauchy-Riemann equations link complex and real analysis. I have completed courses on real and complex analysis, ...
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0answers
39 views

What is Laurent series expansion of $\frac{1}{e^z-1}$ around $z=0$? [duplicate]

Consider $$f(z)=\frac{1}{e^z-1}$$ I want to expand it over $z_0=0$ in a Laurent series. We know that $$e^z=1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots$$ And I know that ...
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34 views

What f(z) will make your 2d self right again? [on hold]

You accidentally got turned inside out while exploring the world of complex functions. What F(z) will make you right again and inverse the "inside out" ? z=x+iy and if I give more info, I will answer ...
2
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2answers
62 views

$f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|\leq 2\}$ such that $\iint_D=|f(z)|^2\,dx\,dy\leq 3\pi$. Maximize $|f''(0)|$

Determine the largest possible value of $|f''(0)|$ when $f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|<2\}$ with the property that $\iint_{D}|f(z)|^2\,dx\,dy\leq 3\pi$. I ...
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1answer
29 views

Schwarz Reflection Principle on a unit disk

Suppose $f$ is a analytic function defined on $\bar{D}(0;1)$ and has real value on the boundary. I'm trying to show $f$ can be extended to entire plane by $$g(z) = \begin{cases}f(z) &, \lvert ...
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7answers
99 views

If $|f| \le |g|$, does analytic continuation of $g$ imply analytic continuation of $f$?

Let $f,g$ be two holomorphic functions on a domain $D$ such that $|f(z)| \le |g(z)|$ for all $z \in D$. Further suppose that there is an analytic continuation of $g$ to a bigger domain $D'$. Does that ...
6
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2answers
36 views

Show that $f$ is continuous at $0$ and it satisfies the Cauchy Riemman conditions but it is not differentiable.

Let $f:\Bbb{C}\to \Bbb{C}$ be defined as $$f(x+iy)= \frac{x^{3}-y^{3}+i(x^{3}+y^{3})}{x^2+y^2} \text{ if} x+iy \neq 0$$ and $f(x+iy)=0$ if $x+iy=0$ Show that $f$ is continuous at ...
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2answers
39 views

Finding a homography in $\hat {\Bbb {C}} $

Let $z_2,z_3,z_4$ be distinct points of $\hat{\Bbb{C}}$. Show that there exists a unique homography $T:\hat{\Bbb{C}} \to \hat{\Bbb{C}}$ of the form $T(z)=\frac{az+b}{cz+d}$ with $ad-bc \neq 0$ such ...
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1answer
34 views

What is the difference between the following $2$ sets?

What is the difference between the following two sets? $\{s\in\mathbb C:\Re(s)\ge1+\delta\},\quad\delta>0$ $\{s\in\mathbb C:\Re(s)>1\}$ I read that $\displaystyle\sum\limits_{n\in\mathbb ...
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0answers
20 views

Integral of ratio of complex polynomials

Let $p(z),q(z) \in \mathbb{C}[z]$ two polynomials with coefficients in $\mathbb{C}$ s.t. $deg(p) = m$, $deg(q) = n$ and $n \ge m +2$. I need to show that $$ \lim_{R \to \infty} \int_{|z| = R} ...
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0answers
26 views

Help with the integral $\int_{0}^{\infty}\frac{x^{y}}{\Gamma(y)}\cos(y)dy$

We have the integral : $$\int_{0}^{\infty}\frac{x^{y}}{\Gamma(y)}\cos(y)dy$$ We have: $$\frac{1}{\Gamma(y)}=\frac{i}{2\pi}\int_{C}(-t)^{-y}e^{-t}dt$$ Where the path $C$ encircling 0 in the positive ...
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0answers
38 views

Possible Connections between Harmonic Analysis, Potential Theory and Analytic Capacity for a Fourier Analyst

So, Folks, here's the deal: After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this ...
2
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1answer
51 views

Question About Filled Julia and Julia Sets

Question: Let $Q_{c}(z) = z^{2} +c $ which $ c \in \mathbb{C}$ and suppose that $z_{0} \in K _{c}$ for the filled Julia Set, $K_{c}$ of $Q_{c}$. Suppose further that $z_{1} = Q_{c}(z_{0})$ and it ...
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1answer
28 views

Equivalence of branched covers of the Riemann sphere

Consider the functions $f(z)=z^4$ and $g(z)=z^4+1$, branched covers of $S^2$. These functions have the same branch data, so they should be equivalent in some way. In what way are they equivalent?
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0answers
10 views

Continuity of the Loewner flow (SLE theory).

In the SLE paper "Basic Properties of the SLE" from Rohde and Schramm, it is mentioned on page 898 that the map $$(y,t)\mapsto g_t^{-1}(iy+\xi(t))$$ is clearly continuous on $y>0,t>0$, where ...
4
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1answer
240 views

Prove that the Mandelbrot Set Is A Closed Set

The Problem: Suppose we define the Mandelbrot Set as the following For $c \in \mathbb{C}$ , $\mathbb{M}$ = $({c:|c| \leq 2}) \cap ({c: |c^2 + c| \leq 2}) \cap ({c: |(c^2+c)^2 +2| \leq 2}) \cap ...
2
votes
2answers
83 views

$\zeta(2n)$ proof [duplicate]

Can anybody pass me on a good source to see the steps in proving, \begin{equation} \zeta(2n) = \frac{(-1)^{k-1}B_2k (2 \pi)^{2k}}{2(2k)!} \end{equation} I know how we start by looking at the product ...
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0answers
27 views

simplifying complex expression

Hi I am trying to simplify the following expression:$$ \left|\frac{1}{a+ib}\left(\frac{J_1(c x)}{J_1(c b)}-x\right)\right|^2,\quad a,b,x\in \mathbb{R}, \ c\in \mathbb{C} $$ Is there a simple way of ...
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3answers
26 views

Limit of a complex valued function.

Let $f(z) = (\frac{z}{\bar{z}})^{2}$ , be a complex valued function , we need to prove that $\lim_{z \to 0} f(z)$ does not exists. So , to prove that its limit doesn't exists , we approach (0,0) from ...
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2answers
85 views

Complex Analysis ( Limits at a point ).

We need to prove that $ \lim_{z \to z_{0}}(z^{2}+c)$ = $z_{0}^{2}+c$ , where c is a complex constant , using $\epsilon - \delta$ definition , where $z , z_{0}$ are complex variables. What I tried : ...
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0answers
37 views

More elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$

Is there a more elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$ than to use Euler's identity and get the general solution through brute computation?
2
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0answers
47 views

isomorphism between function space and complex matrices

How would you show that $\mathcal{L}(X) \cong \mathbb{C}^{n \times n}$, where $X= \mathbb{C}^{n}$. Note that $\mathcal{L}(X)$ denotes the space of linear bounded functions on $X$. Is this a specific ...
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0answers
22 views

Prove the result on connected sets in complex analysis. [on hold]

If $B = S \cup \{$some or all of its limit points$\}$, then $B$ is connected.
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35 views

Prove that a bijective entire function is uniformly continuous

Let $f$ be a bijective entire function. Prove that $f$ is uniformly continuous. I want a direct proof of this without using the fact that $Aut(\Bbb C)$ is the collection of linear polynomials ...
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0answers
16 views

$\int_0^1 \log|x-\zeta|dx\ge (\log|\zeta|+\log|1-\zeta|)/2-1$ [on hold]

I recently came across this inequality: Prove that for any $\zeta\in\mathbb{C}$, $\zeta\ne 0,1$, we have that $$\int_0^1 \log|x-\zeta|dx\ge \frac{\log|\zeta|+\log|1-\zeta|}{2}-1.$$ How do you prove ...
0
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0answers
21 views

curl-free, conservative vector fields in complex analysis

I just verified that for the conjugate of an analytic function $\bar{f}$=u-iv, this conjugate function is curl-free - the Cauchy-Riemann equations forces this to be the case. Then we can consider ...
1
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0answers
38 views

How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
4
votes
2answers
105 views

Geometry of images of maps $f: \mathbb R \to \mathbb C$?

I am having trouble seeing what a continuous map $f: \mathbb R \to \mathbb C$ might look like. If it was linear it would look like a line but it's not clear to me what happens if it's any map. I ...
2
votes
1answer
58 views

Is a bijective entire function uniformly continuous?

Let $f$ be an entire function such that $f$ is bijective. Is then $f$ uniformly continuous? I am thinking on this when trying to compute the analytic automorphisms $Aut(\Bbb C)$. I know that ...
2
votes
1answer
55 views

Difference between line integrals in complex analysis and real analysis,

The formula in complex analysis is $$\int f(\gamma(t))\cdot(\gamma'(t)dt$$ and the formula in the real variable setting, for a gradient field, is: $$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + ...