# Tagged Questions

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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 \left|\frac{T(z_1) - T(z_2)}{1 - \overline{T(z_1)} T(z_2)}\right| = \left|...
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### 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'...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$, 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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### 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 ...
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### 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 ...
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### 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 ...
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}^{\...
### $(B_t)_{t\ge 0}$ 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 ...