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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Existence of such a meromorphic function?

Is there a function $f$ that is holomorphic on $\mathbb{C}-\mathbb{Z} $ and maps into or onto $\mathbb{C}-\mathbb{R}$ ? Into or onto $\mathbb{C}-\mathbb{R}^{+}\cup\{ {0} \}$? All I have been able to ...
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2answers
43 views

On real part of the complex number $(1+i)z^2$

Find the set of points belonging to the coordinate plane $xy$, for which the real part of the complex number $(1+i)z^2$ is positive. My solution:- Lets start with letting $z=r\cdot e^{i\theta}$. ...
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1answer
9 views

How does the author derive this (Difference of analytic functions evaluated at two points)

The conditions are $f:U\to V$ is holomorphic and injective. I basically have 2 questions: Q1) How did the author get $f(z)-f(z_0)=a(z-z_0)^k+G(z)$? Q2) What does "vanishing to order $k+1$" mean? ...
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1answer
101 views

Question regarding a proof that derivatives of injective holomorphic functions are nonzero.

Proposition: Let $\Omega\subseteq\mathbb{C}$ be an open path connected set and suppose $f:\Omega\to\mathbb{C}$ is holomorphic and injective. Then $f'\left(z_{0}\right)\neq0$ for all $z_{0}\in\...
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1answer
51 views

Circles in complex plane.

Find the real value of a for which there is at least one complex number satisfying $|z+4i|=\sqrt{a^2-12a+28}$ and $|z-4\sqrt{3}|\lt a$. My solutions:- Graphical solution:- $|z+4i|=\sqrt{a^2-...
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12 views

find the principal part of the function at its isolated singular point

find the principal part of the function at its isolated singular point, and determine whether that point is a pole, a removable singular point, or an essential singular point. (a) f(z) = z exp(1/z) (...
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1answer
74 views

Prove $f$ is identically zero in $\Omega = \{ z \in \Bbb C:|{\mathop{\rm Re}\nolimits} (z)| < 1,|{\mathop{\rm Im}\nolimits} (z)| < 1\} $

Let $\Omega = \{ z \in \Bbb C:|{\mathop{\rm Re}\nolimits} z| < 1,|{\mathop{\rm Im}\nolimits} z| < 1\} $ and consider the function $f:\bar\Omega\to\Bbb C$ continuous on $\bar\Omega$, analytic in ...
2
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1answer
48 views

Show that for $f$ analytic in $B(0,2)$, $\max_{|z|=1}|\frac{1}{z}-f(z)|\ge 1$?

Let $f:B(0,2)\to \Bbb C$ be an analytic function. Show that $$\max_{|z|=1}\left|\frac{1}{z}-f(z)\right|\ge 1.$$ I tried to write $f(z)$ as power series since it is analytic, it doesn't seem work. I ...
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2answers
490 views

$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$ must be zero and it isn't

I'm trying to evaluate the integral $$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$$ using residue and complex plane integration theory. Let $f(t):=\frac{\sin (t)}{t^4+1}$, $f(z):= \frac{\sin (...
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1answer
52 views

Can I show that $\int_{\gamma(0;r)} \frac{1}{z-a} dz = 0$ when $|a|>r>0$ without using Cauchy Theorem?

I encountered this problem as a previous result of an exercise in a text book way before proving Cauchy Theorem, so I think there must be another way to prove it without it. Show that $\int_{\...
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1answer
18 views

The difference between Taylor and Laurent expansions for Holomorphic functions

I have encountered 2 similar but different theorems on expansions of holomorphic functions to power series, but am not sure how exactly do they differ. Is correct that any $f: U \rightarrow \mathbb{C}...
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3answers
51 views

Cauchy's Integral Formula implies Holomorphicity?

Is the converse direction of the Cauchy Integral Formula true? Meaning, if $f:\mathbb{C}\supseteq U\rightarrow\mathbb{C}$, and $$\forall a \in U \space \space f(a) = \frac{1}{2\pi i}\int_{|z-a|=R}\...
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1answer
38 views

Why is holomorphic function with non-zero derivative a conformal map?

I am new to complex analysis, interested to know why non-zero derivative implies a conformal map. Intuitively, I would think that non-zero derivative means the function is non-constant. Why would ...
1
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1answer
32 views

Proof of the linearity of complex integrals for paths of bounded variation?

I am familiar with the proof of the linearity of complex integrals for piece-wise smooth paths. Nonetheless, complex integrals can be defined for more general paths $\gamma:[a,b]\to\mathbb{C}$ where $\...
1
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1answer
30 views

On the existence of a Möbius transformation

Consider a rational function $f(z)\in\mathbb{R}(z)$ with no poles/zeros on the unit circle $\mathbb{T}=\{z\in\mathbb{C}\,:\, |z|=1\}$. Does there always exist a Möbius transformation $$ \rho\colon \...
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1answer
26 views

Continuity Of Complex Valued Functions

Consider the following complex valued function: $$f(z)=(2+z)Arg(z)$$ Does $f(z)$ have removable discontinuities? (Note: $Arg(z)$ denotes the principal argument.) The following is my approach: We ...
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0answers
11 views

Finding domain where this complex logarithm identity holds

I have not had much exposure to more advanced complex analysis exercises, although this one might be easy, and the solution just completely escapes me. I am determining the set of complex numbers for ...
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1answer
453 views
+50

An entire function of strict order 2

Here is a problem from Stein and Shakarchi Complex Analysis, can somebody help me to solve it? I guess we can use Phragmen-Lindelof theorem but I don't know the exact way. Suppose $f(z)$ is an entire ...
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1answer
24 views

Local normal convergence equivalent to compact normal convergence

Let $X$ be an open subset of $\mathbb{R}^m$ and let $f_n\colon X\to \mathbb{C}$ be complex-valued functions. Then one has the following two notions: $\textbf{1.}$ The series $\sum\limits_{n=0}^{\...
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2answers
34 views

Analytic curve divides disk into two Jordan regions

Let $\gamma:(0,1)\rightarrow\mathbb{C}$ be an analytic Jordan arc. It seems natural to me that for every $\gamma(t_0)$ we can find a disk with center $\gamma(t_0)$ that is divided by $\gamma$ into two ...
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0answers
28 views

Analytic continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Suppose we define a function $f(x)$ by the following sum: $$f(x)= \sum_{n=0}^{\infty} e^{-x E_n}$$ where $E_n$ is a sequence which is at most $O(n)$. It is known $f(x)$ does not form a natural boundry ...
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1k views

Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform

Inspired by this answer, I'm trying to show that $$\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$$ using the inverse Mellin transform. But the answer I get is twice ...
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3answers
27 views

Compact convergence of polynomials

I Want to prove that there is no Sequence of complex polynomial that converges to $f(z)=\frac{1}{z} $ on $D=\mathbb{C} \setminus \{0\}$. Suppose there is a Sequence of complex polynomial converging ...
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0answers
29 views

Operator theory to study a difference equation

I'm not an expert in operator theory (so I'm going to be very informal sorry), but I would like to be given some advice about a problem I have. Let $f$ be a function defined in $C^{\infty}(\mathbb{R})$...
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1answer
30 views

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$ I think I'm supposed to use the following chain of inequalities $$|e^z -1|\leq e^{|z|}-1 \leq |z|e^{|z|}$$ But ...
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0answers
35 views

Geometric Description Of a Set In The Complex Plane

$$S_1=\left\{z:Im\left(\frac{z-z_1}{z-z_2}\right)=0, z_1,z_2 \in \Bbb C\right\}$$ $$S_2=\left\{z:Re\left(\frac{z-z_1}{z-z_2}\right)=0, z_1,z_2 \in \Bbb C\right\}$$ Can someone help me with the ...
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0answers
12 views

A property about the automorphisms of $\mathbb{D}$

I want to prove the next proposition: if $T$ is a Mobius transformation from $\mathbb{D}$ to $\mathbb{D}$, so \begin{equation} \left|\frac{T(z_1) - T(z_2)}{1 - \overline{T(z_1)} T(z_2)}\right| = \left|...
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0answers
12 views

In which context did Klein's $j$-invariant first appear?

I'm currently studying Klein's $j$-invariant; while all books I considered put emphasis on the unexpected connections it has with other fields of mathematics (e.g. the "monstrous moonshine"), I couldn'...
0
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1answer
33 views

Determining if a function is harmonic in a fast way

Determine which function is harmonic in $\mathbb R^2$: $$\text{a) } y^2 \qquad \text{b) }x^2 + y^2\qquad \text{c) } e^x\qquad \text{d) }\operatorname{Im}((x + iy)^5)$$ I had this question come up ...
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0answers
12 views

Argument principle for meromorphic forms on Riemann surfaces

Let $X$ be a compact Riemann surface and $D \subset X$ a compact domain with boundary $\partial D$. Let $\omega$ be a meromorphic $1$-form in a neighborhood of $D$ which does not have neither zeros, ...
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1answer
23 views

Show $|f(z)|<3$ for each $z\in B(0,1)$?

If $f:B(0,2)\to\Bbb C$ be an analytic function satisfying $|f(z)-2|<1$ for each $z\in \Bbb C$ such that $|z|=1$. show that (a) $|f(z)|<3$ for each $z\in B(0,1)$? (b) $f(z)\neq0$ for each $z\...
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1answer
26 views

Existence of biholomorphic map from unit disk to itself that interpolates one set of points

How do you prove that given two points $z_{1}, w_{1} \in D = \{z: |z|<1\} $, there exists a biholomorphic (bijective and analytic) function $f: D \to D$ such that $f(z_{1}) = w_{1}$? Perhaps using ...
2
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1answer
104 views

Compute the sum of the series.

I just see the equality in my textbook, but I really have no idea how it arises (maybe it is obvious to the author), and it seems Fourier methods are not applicable. I would appreciate if someone ...
1
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1answer
75 views

Are all the zeros of $1-a_2x^2+a_4x^4-a_6x^6+\cdots$ real for $a_{2n}>a_{2(n+1)}$ with $a_{2n+1}=0$ and $a_{2n}>0$?

This question is related to a previous question of mine. I was not pleased about the conditions I provided there. I had something different in mind but I failed in stating it. So here are the ...
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1answer
27 views

How does one prove that two punctured disks are conformally equivalent? [on hold]

Let D1 = {z: 0 < |z| < R1} and let D2 = {z: 0 <|z| < R2}. Prove that D1 and D2 are conformally equivalent.
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1answer
59 views

Help on line integral $\int\limits_\gamma \frac{1}{(z + 1)(z + 2) \cdot \ldots \cdot (z + r)} dz$

I need help on the following line integral: $$\int\limits_\gamma f dz = \int\limits_\gamma \frac{1}{g} dz = \int\limits_\gamma \frac{1}{(z + 1)(z + 2) \cdot \ldots \cdot (z + r)} dz$$ For a fix $r \in ...
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1answer
30 views

bounded components of the intersection of two planar domains

It seems to be intuitively clear that if U is a domain in the plane having a bounded complementary component C, then C is also a complementary component of the intersection of U with an open disk D ...
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1answer
228 views

Arnold's proof of Abel's theorem

I'm seeking help understanding this video. The author considers the equation $ax^5+bx^4+cx^3+dx^2+ex+f = 0$ and shows both the coefficients $a, b$... and solutions $x_1, x_2$... in the complex ...
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1answer
63 views

Are derivatives actually bounded?

Suppose you a function $f$ which is differentiable, with the property that $$ f^{(n)} (0) = (n!)^2 $$ And in general $$ f^{(n)} (a) = O((n!)^2)$$ For any $a \in \mathbb{R}$. This function ...
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1answer
38 views

Prove that $u\leq v$ everywhere.

Let $u$ be a subharmonic function on an open set $U$ in $\mathbb{C}$, and let $v$ be an upper semicontinuous function on $U$ such that $u\leq v$ almost everywhere. Prove that $u\leq v$ everywhere. ...
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10 views

Gelfand spectrum of $l^1(\mathbb Z)$ is homeomorphic to $\mathbb T =\{z \in \mathbb C : |z|=1\}$ [on hold]

Show that the Gelfand spectrum of $l^1(\mathbb Z)$ is homeomorphic to $\mathbb T =\{z \in \mathbb C : |z|=1\}$. Please provide some hints. Thanks in advance.
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3answers
2k views

Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$)

Most of us are aware of the famous "Basel Problem": $$\sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$ I remember reading an elegant proof for this using complex numbers to help find the value of ...
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1answer
44 views

$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$, for complex variable $z$.

I want to find this limit for complex variable $z$ $$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$$ In the real case I know $\sin(z)$ is bounded by $-1, 1,$ and the limit is $0$. But in the complex case ...
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2answers
41 views

Complex Frequency Shifting in Fourier Transform

When dealing with Fourier transforms, it is often useful to take advantage of the following property in order to simplify work: $$\mathcal{F}(e^{i\omega_0t}f(t))=G(\omega-\omega_0)$$ where $G(\omega)...
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1answer
29 views

complex modulus and square root

I am failing to understand something about complex square roots: If we fix the argument $\theta\in(0,2\pi],$ that is we take the positive real line as branch cut, than for $z=r\mathrm{e}^{i\theta}$, $...
3
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2answers
37 views

Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$

I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also ...
26
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2answers
353 views

If $f$ is a smooth real valued function on real line such that $f'(0)=1$ and $|f^{(n)} (x)|$ is uniformly bounded by $1$ , then $f(x)=\sin x$?

Let $f : \mathbb R \to \mathbb R$ be a smooth ( infinitely differentiable everywhere ) function such that $f '(0)=1$ and $|f^{(n)} (x)| \le 1 , \forall x \in \mathbb R , \forall n \ge 0$ ( as usual ...
0
votes
1answer
31 views

An application of the open mapping theorem

Let $U\subseteq \mathbb C$ be a domain and $a,b,c \in \mathbb R$ with $a^2+b^2>0$. Determine all on $U$ holomorphic functions $f$ which satisfy: $a\cdot Re(f) + b\cdot Im(f) +c = 0$. I ...
0
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0answers
29 views

The complex version of the Riemann-Lebesgue lemma

I can't prove the complex version of the Riemann-Lebesgue lemma. $$ f(x) \in \mathbf{C} \\ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos nx \, dx, \quad b_n = \frac{1}{\pi} \int_{-\pi}^{\...
0
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0answers
34 views

$(B_t)_{t\ge 0}$ be Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t}$ is an analytic function.

Let $(B_t)_{t\ge 0}$ be a one-dimensional Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t} \; \text{for all} \; t\ge 0, \xi \in \mathbb{R}$ is an analytic function. A more general question ...