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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21 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 ...
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20 views

Can anyone help me with this complex-valued integral?

$$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ 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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14 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
39 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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22 views

Complex Variables 1 (Montel's thrm

given $0 < c \in R$, there exist $\epsilon > 0 $ such that $a_k \in C$ & $\sum_{k=1}^{\infty}|a_k| \leq c$ implies $\sup_{\frac{1}{2}\leq x\leq1}|1 - \sum_{k=1}^{\infty}a_k x^k| \geq ...
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22 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 $f(1)=1$, ...
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20 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 ...
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24 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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22 views

Best way to find Residue? [on hold]

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!
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31 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
27 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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23 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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22 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 ...
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1answer
25 views

Determining whether a continuous function is harmonic

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
65 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 ...
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1answer
32 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 ...
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1answer
16 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 ...
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1answer
17 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 $\log z$ ...
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1answer
80 views

A UCLA Qualifying Complex Analyis Problem , possibly related to Phragmén-Lindelöf Theorem

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 $f(z) \to 0, z \to 0$ ...
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28 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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2answers
21 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 ...
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1answer
31 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 = ...
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0answers
15 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 ...
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26 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 ...
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1answer
41 views

How can I expand this

How can I expand $\dfrac{\pi \csc(z\pi)}{(2z+1)^3}$? so then I can find the residue ? thanks
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24 views

Existence of analytic function on a unit disc $\triangle$ [on hold]

Let $\triangle$ be the open unit disc. Then can there be analytic functions with the property (1) $f(\frac{3}{4})=\frac{3}{4}$ and $f'(\frac{2}{3})=3/4$ 2) $f(\frac{3}{4})= -\frac{3}{4}$ and ...
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17 views

Prove that the no of solutions of the equation $f(z)=w$, counted with Multiplicities for $w$ varying in $D_2$, is constant on $D_2$

Prove that the no of solutions of the equation $f(z)=w$, counted with Multiplicities for $w$ varying in $D_2$, is constant on $D_2$. Def: A map $f\colon X \to Y$ is said to be proper if ...
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1answer
18 views

Understanding a Wermer's counterexample.

I am reading some lecture notes on holomorphic functions of several complex variables, see page 105. The part I am struggling with is a proof by Wermer I have asked about runge domains, and ...
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1answer
26 views

Why the function $w=e^z$ maps the lines $x=c$

The questions asks: Explain why the function $w=e^x$ a) maps the lines $x=c$, with $c$ a constant, onto the circles $w=e^c$ b)maps the lines $y=c$, with $c$ a constant, onto half rays $\theta=c$ ...
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0answers
18 views

Winding number and homotopic paths (Complex Analysis)

I know that if $\Omega$ is open subset of $\mathbb{C}$ and $\gamma_1$,$\gamma_2$ are two homotopic paths such that $\gamma^*_1$,$\gamma^*_2 \subset \Omega$ then $\forall \alpha$$\in$ ...
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1answer
33 views

Identify regions where $\sin(e^x)$ is analytic

The question asks to differentiate $\sin(e^x)$ and then determine where it is analytic. I know how to differentiate it to get $f'(z) = e^z \cos(e^z)$ but I am unsure how to find where it is analytic. ...
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1answer
20 views

Singularities and Residue

For part (a) the singularity is 1/root2 + i/root2 ? And it is a pole of order 1? I am having trouble calculating the residue So far I have: residue = limit (as z tends to 1/root2 + i/root2) of ...
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61 views
1
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1answer
38 views

Residue of essential singularity

$$f(z)=\sin(z)e^{1/z}$$Find the residue of $f$ at $0$. I think there is an essential singularity at $z=0$ ? How do I compute the residue of this... I know how to compute the residue of poles but not ...
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24 views

Can anyone prove this using residue theorem? [on hold]

Can anyone prove this using residue theorem? $$\sum\limits_{k=0}^\infty \frac{(-1)^k}{(2k+1)^3}=1-\frac{1}{27}+\frac{1}{125}-\dots=\frac{\pi^3}{32}$$
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1answer
32 views

Dual plot for complex roots of quadratic equation

Real roots of quadratic equation $ x^2 - \sqrt 3 x + 1/2 =0 \tag{1} $ can be plotted on $x$- axis as its parabola intersection at $ (\sqrt 3/2 \pm 1/2,0). $ In an improvization I assign ...
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2answers
68 views

The value of $\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2}$ on $\mathbb{C}\setminus\mathbb{Z}$

Show, for $z\not\in\mathbb{Z}$, that $$\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2} = \frac{-\pi}{z\tan(\pi z)}$$ Hint: You may assume that there exists $C$ such that $|\pi\cot(\pi w)|\leq ...
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70 views

What is reflection across parabola?

Reflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but at ...
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1answer
30 views

Relation between runge domain and polynomial convexity

Are these concepts the same? Just to state the definitions Definition 1 A domain $\Omega \in \mathbb{C}^n$ is a Runge domain if every function $f \in H(\Omega)$ can be approximated, uniformly on ...
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2answers
43 views

Apply the Cauchy-Goursat theorem to show that $\int_C \operatorname{Log}(z+2)\, dz=0$ on a unit circle.

Cauchy-Goursat theorem. If a function $f$ is analytic at all points interior to and on a simple closed contour $C$, then $$\int_C f(z) \,dz=0.$$ This is a problem from Churchill's Complex Variables. ...
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1answer
28 views

Differentiability on $Re(z)^2$

Suppose that $l(z)=Re(z)^2$. Is $l$ differentiable at $z_0 \in \mathbb{C}$? What is $f'(z_0)$? Where is $l$ differentiable? Write $w=u+iv$ and $z_0=x_0 +iy_0$. So $$\lim \limits_{w \rightarrow 0} ...
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1answer
25 views

Find and classify singularities

Let $f(z)=\frac{e^{-z}\sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$. Find and classify the singularities of $f(z)$. So far I have that there are singularities at $1$, $2$ and $-2$ $2$ and $-2$ are simple ...
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2answers
36 views

Find and classify the singularities of $f(z)$

Let $$f(z) = \frac{e^z\sin(3z)}{(z^2-2)z^2}$$ Find and classify the singularities of $f(z)$. So far I have that there are singularities at $0$, $\sqrt{2}$ and $-\sqrt{2}$. Are these correct? ...
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1answer
23 views

Boundary preserving map

Let $K\subseteq\mathbb{R}^2$ be a compact set. Is it true that for a continuous map $p:K\to\mathbb{R}^2$ we have: $p(\partial K)=\partial p(K)$? Are there any generalizations? P.S. Note that ...
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1answer
24 views

Runge's Theorem for meromrophic functions

Is there a name for this extension of Runge's theorem? Theorem: Let $K\subset\mathbb{C}$ be compact, and let $A\subset K^c$ be a set which intersects each component of $K^c$. Let $f$ be meromorphic ...
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44 views

Multivalued Functions for Dummies

I have been studying complex analysis for a while, but I still cannot "get" how multivalued functions work. Despite having it explained to me many times, my brain cannot process it. For example, I ...
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27 views

Complex limit proof

Definition: Say $f: S \longleftrightarrow \mathbb{C} \longleftrightarrow \mathbb{R}^2$, $z_0 \in S$, $l \in \mathbb{C}$. We say $$\lim \limits_{z \rightarrow z_0} f(z) = l_0$$ if $\forall \varepsilon ...
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1answer
77 views

Proving an entire function is constant

I'm trying to prove that the entire functions such that \begin{equation*} n^2f(1/n)^3+f(1/n)=0 \end{equation*} for all $n\in\mathbb{N}$, are constant. I suppose I should prove that $f$ is bounded ...
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1answer
19 views

Derivative of a complex conjugate

I anticipate that this is a stupid question, but suppose $c \in C$. What is $\frac{\partial c^{*}}{\partial c}$? I've been trying and failing for about an hour to figure it out from the definition of ...
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2answers
55 views

problems with singularity $0$ of $\int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz$.

I have the complex integral \begin{equation*} \int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz \end{equation*} where $W$ is a circle with radius $6$ and centered at $0$. Obviously we have two ...