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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24 views

Let $f\in H(\mathbb{C})$. Prove that: $\exists_{M\in\mathbb{R}^+} \forall_{z\in\mathbb{C}}\ \ \ \ |f(z)|> M \Rightarrow f$ is constant

Let $f\in H(\mathbb{C})$. Prove that: $\exists_{M\in\mathbb{R}^+} \forall_{z\in\mathbb{C}}\ \ \ \ |f(z)|> M \Rightarrow |f(z)|> M \Rightarrow f$ is constant Completely don't know how to bite ...
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0answers
39 views

Laurent series confusion

I've split it up into partial fractions and got $1/z$ - $2/(z-1)$ + $1/(z-2)$ but I'm unsure sure what to do now. I think I have done part $(i)$. I get $$z^{-1} + \sum_{n=0}^\infty ...
1
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1answer
72 views

How find the poles/residues of $\int_{-\infty}^\infty \frac{x^2 \, dx}{1 + x^4}$

I'm trying to find the poles/residues of this integral: $$\int_{-\infty}^\infty \frac{x^2 \, dx}{1 + x^4}$$ I've been given this attempt for a solution, but I don't really understand the procedure ...
0
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1answer
15 views

Question about a certain step in Rudin's General Cauchy Theorem proof

I am having trouble seeing a certain claim that Rudin makes in proving his "Global Cauchy's Theorem": $\textbf{Cauchy's Theorem.}$ Suppose $f$ is holomorphic in $\Omega$, which is an open set in ...
2
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1answer
38 views

Laurent series of $1/({z^3-z})$

Question: Find the Laurent series of the function $$f(z) = \frac{1}{z^3 - z}$$ at the domain $|z-1|>2$. Attempt: So we have $$\frac{1}{z(z-1)(z+1)}$$ and we only have to find a Laurent ...
3
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1answer
44 views

Show that an entire function is a proper if and only if it is a nonconstant polynomial

Show that an entire function (Holomorphic on $ \mathbb C$) is proper if and only if it is a non constant polynomial. Def:A map $f:X\to Y$ is called proper if $f^{-1}(K)$ is compact for every ...
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0answers
46 views

How does $\cos (2z) = e^{2zi}$?

In my notes, they are solving $$\int \limits_{- \infty}^{\infty} \frac{\cos(2x)}{x^2 +1}$$ and they let $$f(z) =\frac{e^{2zi}}{z^2 +1}$$ but how did the numerator become that? I wrote it as $\cos(2z)$ ...
1
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1answer
56 views

suppose $f(x)$ is an entire function and everywhere $|f'(z)| \leq |z^2+1|$ and further $f(0) = f'(0) = 1$. Determine $f$

Suppose $f(z)$ is an entire function and everywhere $|f'(z)| \leq |z^2+1|$ and further $f(0) = f'(0) = 1$. Determine $f$. I tried using Liouville's theorem but i don't know if $f'(z)$ is an entire ...
1
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1answer
25 views

About Fourier transform and complex conjugate

why this passage is correct ? \begin{equation*} \mathscr{F}[h(-\tau)] = H^*(f), \end{equation*} when $h(\tau)$ is a real function of real variable $\tau$, and $H^*(f)$ is the complex conjugate of ...
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0answers
29 views

Contour integration, cos(z)sin(z)

Evaluate \begin{equation*} \int_{\Gamma}\cos(z)\sin(z)dz,~\Gamma:\gamma(t):=\pi t+(1-t)i,~0\leq t\leq 1. \end{equation*} I think I should do it using this \begin{equation*} ...
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0answers
37 views

Is $\overline{z} $ independent of $z $?

Ahlfors' Complex Analysis says the following: $x=1/2(z+\overline {z} )$ and $y=-1/2i (z-\overline {z}) $. Hence, for a function $f (x, y) $, we have $\frac {\partial f}{\partial z}=1/2(\frac ...
0
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0answers
25 views

Show that $f(V_0)\cap f(V_1)\neq\emptyset$

Let $U$ be a connected subset of $\mathbb C$ and $z_0,z_1\in U$ and if $f$ is holomorphic on $U\setminus\{z_0\}$, with essential singularity in $z_0$, prove that for any subsets $V_0,V_1$ of $U$ ...
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0answers
31 views

Algebraic approach to Local Analytic Complex Geometry

I'm attending a second course in Complex Analysis from a geometrical point of view. In the final part of the course we have discussed about germs of complex analytic sets and their algebraic ...
5
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2answers
34 views

let $f$ be holomorphic on the unit sphere and $|f(z)| = 1$ for $|z| = 1$ and $f(-1) = 1$. Furthermore $f$ has no zero's, determine $f$

let $f$ be holomorphic on the unit sphere and continous on the closure, suppose $|f(z)| = 1$ for $|z| = 1$ and $f(-1) = 1$. furthermore $f$ has no zero's, determine $f$. So far i know with the ...
1
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1answer
16 views

Interpolation with nonvanishing constraint

Let $x_1,x_2,\ldots,x_n$ be distinct complex numbers. Let $y_1,y_2,\ldots,y_n$ be nonzero complex numbers, and let $K$ be a bounded subset of $\mathbb C$. Does there always exist a polynomial $P$ such ...
0
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1answer
18 views

Differentiation or integration term by term of the Laurent series!

Let $f(z)$ be an analytic function in the annual $r< |z|<R.$ Then $f(z)$ has the Laurent expansion series in this annual. My question is that: Can we derivative (or integrate) term by term from ...
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1answer
24 views

Find value of complex function at a point

Let $f(z)$ be analytic in $ D = \{z \in \Bbb C : |z| < 1\}$, and $f(z) = 1$ when $Im(z) = 0$ and $-\frac{1}{2} \leq Re(z) \leq \frac{1}{2}$. What is the value of $f(\frac{1}{2}+i\frac{1}{2})$? I'm ...
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0answers
36 views

Inequality on complex polynomial

For every $a\geq 0$, let $p_a(z)=1-z+az^3$. What is the maximal value of $a$ such that $$ p_a(|z|)\leq |p_a(z)| $$ for all $z\in \mathbb C$? EDIT: I claim that $a=\frac{4}{27}$ is the maximal value. ...
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2answers
33 views

Taylor expansion of a complex function

Trying to find Taylor series of $$\frac{z^2}{(1+z)^2}$$ I write it in the form $1- \frac{2}{1+z} + \frac{1}{(1+z)^2}$ and I can find Taylor expansion for each factor, is there another method without ...
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2answers
40 views

Taylor series of $1\over z^2$

How to find the Taylor series of $1\over z^2$ near $2$ ( in the power of $z-2$) I have tried to write it in the form: $1\over ((z-2)^2+4z-4)$ But I reached nothing, any help please
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0answers
15 views

Meromorphic complex function [closed]

I want to find all function $f$ which are meromorphic in $C$ and satisfy $|f(z) - tan(z)| < 2$ for all $z$ which are neither poles of $f$ nor poles of $tan(z)$
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0answers
26 views

calculate complex integral $\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$

I don't know how to calculate this complex integral: $$\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$$
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1answer
42 views

Show that $f$ is constant in $D(0,1)$. [closed]

Considere this, $f: D(0,1) \to D(0,1)$ analytic. Suposse that $|f(z^2)|>|f(z)|$, for all $z \in D(0,1)$. Show that $f$ is constant in $D(0,1)$. Any help.
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1answer
28 views

Taylor series in complex analysis [closed]

I am working on finding the Taylor series of $$\frac1{az+b}$$ in powers of $z.$ How to start with it Any help in details...
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1answer
22 views

Residue Calculus - Showing that the quotient of polynomials have integral $0$ in a simple closed contour in a special case.

I'm having difficulty understanding the solution to the following problem. In the solution below, I can't understand why since $b_m\neq 0$, the quotient of these polynomials is represented by a ...
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0answers
22 views

Exactly one supporting line for a $C^1$ Jordan curve [on hold]

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a convex Jordan curve (closed, simple, continuous) that has $C^1$ regularity, with $\gamma '(t)\neq 0,\ \forall t\in [a,b]$. Prove that there is exactly one ...
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0answers
38 views

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I ...
2
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1answer
47 views

Gaussian integral with a shift in the complex plane

$$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ where $a\in \mathbb{R}$. I don't know where to start but have reasons to believe the answer is $\sqrt{\pi}$. Namely $\int_{-\infty}^\infty ...
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1answer
26 views

Antiholomorphic function

Let f be an antiholomorphic function in C. $z_0 \in C - C(0,1). $ Show that $\frac{1}{2 \pi i}\oint \frac {f(z)}{z-z_0} = \begin{cases}f(0) &\text{for } |z_0| < 1\\f(0) - f(\frac{1}{z_0}) ...
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1answer
55 views

Why is $|e^{i \lambda z}| |e^{- \lambda y}|= |e^{- \lambda y}|$ here?

Let $z \in \Gamma (R)$ where this is the upper semi circle centred at the origin with radius $R>1$. Let $z=x+iy$ with $x \in \mathbb{R}$ and $y \geq 0$. So $$|e^{i \lambda z}|=|e^{i \lambda z}| ...
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2answers
53 views

A question related to Montel's Theorem

Given $c>0$, there exist $\varepsilon > 0,$ such that, whenever $\{a_n\} \subset \mathbb C$ and $\sum_{n=1}^{\infty}\lvert a_n\rvert \le c\,$ implies that $$ \sup_{\frac{1}{2}\leq x\leq1}\left|1 ...
0
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1answer
23 views

Show that if $f(z)=\frac{\operatorname{Log}z}{z-1}$ when $z\neq 1$ and $f(1)=1$, then $f$ is analytic throughout the domain.

$\operatorname{Log}z=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(z-1)^n \; (|z-1|\lt 1).$ Use this fact to show that if $$f(z)=\frac{\operatorname{Log}z}{z-1} \; \text{when} z\neq 1$$ and $f(1)=1$, ...
2
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1answer
38 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
0
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1answer
33 views

With the aid of series, show that if $f(z)=\frac{\operatorname{cos}z}{z^2-(\pi/2)^2}$, then $f$ is an entire function.

Prove that if $$f(z)= \begin{cases} \frac{\operatorname{cos}z}{z^2-(\pi/2)^2}, & \text{when} \; z\neq \pm \pi/2, \\ -\frac{1}{\pi}, & \text{when} \; z=\pm \pi/2, \end{cases} $$ then $f$ is ...
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0answers
42 views

Best way to find Residue? [closed]

I know that this is a strange question to ask on this website but I am dying to know a method that you can always use to find the residue of any complex function. Please help! We learned three types ...
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0answers
43 views

Laurent series for $\frac{2}{(z)(z-1)(z-2)}$

! So I think I am getting the hang of Laurent Series, but having a bit of trouble with one of the fractions for part a). So I split this up in to partial fractions: $\frac{1}{z} - ...
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2answers
42 views

If $f(z)= \frac 1z $ be defined and analytic on region $ |z| \gt 1 $ in $ \Bbb C $ then can we find an entire function $g$ such that :

$g$ should be such that $f(z)=g(z)$ on $ |z| \gt 1$ in $\Bbb C $. Now,Can we plainly apply uniqueness theorem and say that such a function $g$ can not exist?
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0answers
30 views

Contour integral from first principles

What does it mean by 'evaluate from first principles? Does it mean use ? For part (a) do I parametrise as $\gamma(t)=a+2e^{it}$ with $t$ between $0$ and $2\pi$? Doing this I end up with the ...
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0answers
25 views

Counting poles that are shared between $f$ and $g$

Suppose I have a meromorphic function $f(z)$ with poles at $f_i$ and $\mathcal{Res}(f,f_i)=1$, and $g(z)$ with poles at $g_i$ and $\mathcal{Res}(g,g_i)=1$. I would like to construct a function ...
3
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1answer
44 views

Does a weaker form of the mean value property already imply harmonicity for continuous functions?

If $u:\mathbb{C}\to \mathbb{R}$ is continuous and satisfies $u(z)=\frac{1}{2\pi}\int_0 ^{2\pi}u(z+\frac{e^{i\theta}}{n})d\theta$ for all $n\in \mathbb{N}$ and $z\in \mathbb{C}$, is $u$ harmonic? What ...
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3answers
66 views

Finding local max of analytic function

Given a function $f=z^2+iz+3-i$. I need to find the the maximum of $|f(z)|$ in the domain $|z|\leq 1$ I know that the maximimum should be on $|z|=1$ but when I tried to put $z=e^{i\theta} $ in the ...
1
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1answer
46 views

Contour integral $|z-i|=1/9$

Calculate \begin{equation*} \int_{\Gamma}\frac{1}{z^4+16}dz, \end{equation*} where $\Gamma :|z-i|=\frac{1}{9}$. I have asked I similar question to this but I still do not understand.... when I find ...
2
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2answers
25 views

Is any simply connected domain conformally equivalent to Cartesian product of unit disks?

By Riemann mapping theorem, any simply connected domain is conformally equivalent to the unit disk. Is any simply connected domain in the complex plane conformally equivalent to the Cartesian product ...
2
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1answer
18 views

If an entire function $f$ satisfies $|f(z)| \le |\log z|,$ what can we say about $f$?

Let $f$ be an entire function. Define $\Omega=\mathbb{C}-(-\infty,0]$, the complex plane with the ray $(-\infty,0]$ removed. Suppose that for all $z \in \Omega$ , $|f(z)| \le |\log z|$, where $\log z$ ...
3
votes
1answer
112 views

A UCLA Qualifying Complex Analyis Problem , possibly related to Phragmén-Lindelöf Theorem [on hold]

Let $f$ be a bounded analytic function on the open right half plane such that $f(x) \to 0, x\to 0$ along the positive real axis. Suppose $0<\phi<\pi/2$. Prove that $f(z) \to 0, z \to 0$ ...
5
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1answer
47 views

Show $f(z)$ can be analytically continued and $F(z+4)=F(z)$ for resulting entire function

I'm working on some past qualifying exam problems in complex analysis and I'm quite stuck on this one: Let $f(z)$ be analytic in $\{z\in\mathbb{C}\,:\,|\text{Re }z|<1\}$ and continuous on the ...
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votes
2answers
23 views

equation of a line into the complex form

So if i am given an equation of a line in complex form for example $Re|(1+i)z| = 0$, I could turn this into its real counter part on the x-y plane and graph it. Is there a way to go in the other ...
0
votes
1answer
32 views

Write $1/z$ as a power series

Show that the function $f(z)=1/z$ can be represented as a power series in a ball $B(z_0,r)$, where $z_0 \neq 0$. Find the radius of convergence of this power series. $$f(z)=\frac1z = ...
0
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0answers
21 views

Inverse of a constant function on an open set

I was working on holomorphic functions and Riemann surfaces, and I was wondering about the inverse of a constant function: Let $f:U\rightarrow V$ be a holomorphic function between two Riemann ...
0
votes
0answers
27 views

complex variable inequality

Let $B$ and $C$ be nonegative real numbers and $A$ a complex number. Suppose that $$ 0\leq B-2Re(\overline{\lambda}A) + |\lambda|^2 C \ \forall \ \lambda \in \mathbb{C} $$ Conclude that $|A|^2 \leq ...