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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A UCLA Qualifying Complex Analyis Problem , possibly related to Phragmen-Lindelof 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 $f(z) \to 0, z \to 0$ uniformly ...
3
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1answer
25 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
20 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
30 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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13 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
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24 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
32 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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0answers
23 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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1answer
15 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
16 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
25 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 $\theta=c$ ...
0
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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$ ...
0
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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
19 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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0answers
51 views
1
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1answer
36 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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0answers
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
26 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 ...
3
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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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61 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 ...
2
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1answer
29 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 ...
2
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2answers
41 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
24 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
34 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
22 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 ...
2
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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 ...
4
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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 ...
2
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1answer
75 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 ...
0
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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
53 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 ...
3
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1answer
16 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
0
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1answer
36 views

Existence of analytic function on a unit dsic (Converse of Schwaraz Pick Lemma )

(Schwarz - Pick Lemma) Suppose that $f$ is analytic on the Unit Disk $\triangle$ and satisfies the following two conditions : (1) $|f(z) \leq 1$ for all z $\in \triangle$ (2) $f (a) = b $ for some ...
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1answer
15 views

Contour integral of non analytic exponential function

The value of the integration of the function $f(z)$ over the circle of radius 3 centered at $z=1$, where $f(z) = e^{\frac{-1}{(z-1)^2}}$ this function has a pole at $z=1$ of $2$nd order. I don't ...
4
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2answers
74 views

Is entire function a polynomial? [duplicate]

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function, and suppose that for every $z\in \mathbb{C}$ there exists $n_z\in \mathbb{N}$ such that $f^{(n_z)}(z)=0$. Is $f$ necessarily a polynomial?
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1answer
11 views

Classification of conformally equivalent annuli via periods

How does one define the periods that appear in this question and show they are conformally equivalent? Are the details worked out in a textbook somewhere? Presumably we do something like take the ...
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1answer
17 views

Normal Family complex

I search about a family of holomorphic complex function that not normal but their derivative is normal. the definition of normality similar to Montel's Thrm
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1answer
31 views

Given $z$ show that $\left | z\right | = 2\sin\theta$ and $\arg z = \theta$

I've been attempting this complex-related question but couldn't quite crack the challenge. (b) Given that $$z={1-\cos 4\theta+i\sin 4\theta\over\sin 2\theta+2i\cos^2\theta},$$ show that ...
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1answer
16 views

Complex Analysis Triangular Inequality

I recently started learning Complex Analysis and as of now don't have much command over it, I am stuck up with this assignment Question of mine which is as follows: If $|Z_i| < 1$ and $V_i ≥ 0$ ...
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25 views

Complex Analysis Modulus Property

My teacher has asked an extra question apart from the one mentioned below in the picture: What will happen if |z| = |w| = 1 ? With respect to following answer: I don't see any point in that ...
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1answer
14 views

Orthogonal complex matrices: polar decomposition

Is there a decomposition of $SL_n(\mathbb C)$ as a product of $O_n(\mathbb C)\times Sym_n(\mathbb C)$ ? I mean is there a result as the polar decomposition but with orthogonal (not unitary)? thanks ...
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1answer
47 views

Integration of hyperbolic functions. [on hold]

Kindly solve this integral. I shall be very grateful. $$ \int_0^{\infty}\frac{\mathrm{e}^{-x}}{\mathrm{e}^{ax}-1}dx $$ Thanks.
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1answer
48 views

Computing $\int_{\partial S} \frac{1}{1+z^n} dz$

Let $S=\{re^{it} : 0<r<R, 0< \varphi < 2\pi/n\}$ for some $R>1$ and $n\geq 2$. How can we compute $$\int_{\partial S} \frac{1}{1+z^n} dz?$$ I can't compute it directly, so I assume I ...
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0answers
25 views

Beyond Schwarz Lemma

Let $f(z)=a_1z+a_2z^2+a_3z^3... $ be a Schwarz function then by lemma $|a_1|\leq 1 $.But what is known about the higher coefficients? For example ; what can be said about $ max [a_1+a_2]$ ? Is there ...
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1answer
17 views

Establishing a Variant of the Mean Value Property of Harmonic Functions

Let $u:U\to \mathbb{C}$ be harmonic and $\overline{D}(P,r)\subset U$. Verify the following variant of the mean value property of harmonic functions: $$u(P)=\frac{1}{2\pi r}\int_{\partial ...
3
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3answers
40 views

Radius of convergence of power series of complex $\log$

Let $f(z) = \log(z)$ for $z\in \Bbb{C}\setminus (-\infty,0]$. Since $f$ is holomorphic on its domain, we know it has a power series development about each point $z_0\in \Bbb{C}\setminus (-\infty,0]$. ...
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0answers
35 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
2
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1answer
33 views

Non-constant entire function-bounded or not? [duplicate]

Show that if $f$ is a non-constant entire function,it cannot satisfy the condition: $$f(z)=f(z+1)=f(z+i)$$ My line of argument so far is based on Liouville's theorem that states that every bounded ...
3
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1answer
45 views

A consequence of Schwarz lemma

Suppose that for some $\epsilon>0$ the function $f$ is holomorphic on $B(0,1+\epsilon)$ such that $f(a) = 0$ and $|f(z)|\leq1$ if $|z| \leq 1$. Prove for $|z| \leq 1$: $$|f(z)|\leq ...