0
votes
0answers
15 views

How to write the condition for Image of a function?

If $\Omega_l$ is $\Omega$ with $|x|<l$ and if $\Omega_S$ is the image of $z$ under mapping how we will write the condition for it. Am I right if I write $\Omega_S$ is $\Omega$ with $|S|<l$ or ...
0
votes
1answer
61 views

Is there a name for the one-point compactification of $\mathbb{C}$?

Let $\hat{\mathbb{C}}$ be the one-point compactification of $\mathbb{C}$. This space $\hat{\mathbb{C}}$ is called the Riemann sphere. If I want to designate the topology $\tau$ on ...
1
vote
1answer
66 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
1
vote
1answer
34 views

If $\zeta$ is a function of characters what does it mean for it to be regular?

This is from lemma 2.4.1 of Tate's thesis. Lemma 2.4.1: A $\zeta$-function is regular in the "domain" of all quasi-characters of exponent greater than $0$. proof: We must show that for each ...
0
votes
0answers
24 views

How do i visualize plots illustrating complex functions?

Here's an example : http://en.m.wikipedia.org/wiki/Essential_singularity What is this colored diagram illustrating complex function called? And how should i understand this diagram? What do colors ...
3
votes
0answers
45 views

In complex analysis, is there a special name for functions that can be written as an infinite product of linear factors?

Let $Z$ denote a subset of $\mathbb{C}$. Then some functions $f : Z \rightarrow \mathbb{C}$ have the property that there exist sequences $a,b : \mathbb{N} \rightarrow \mathbb{C}$ such that for all $z ...
0
votes
1answer
24 views

Complex function terminology question

Suppose that I have a function $f:\mathbb{C}\rightarrow\mathbb{C}$. Representing the complex number in polar notation $z=re^{i\theta}$, I integrate the phase $\theta$ out as follows: ...
1
vote
4answers
114 views

Are there two conventional definitions of “holomorphic”?

In Walter Rudin's Real and Complex Analysis, second edition, on page 213, two definitions are stated. One of them says the derivative of $f$ at $z_0$ is $$f'(z_0)=\lim_{z\to ...
2
votes
1answer
178 views

please help me grasp the literal meaning of residue

In complex analysis we study a term RESIDUE of a function given by some formulas. While going through its meaning I found that it means left out term or remainder kind of thing. so I was wondering why ...
2
votes
2answers
74 views

Please help me check my metric definition of isolated point

I translated the word definitions into the more symbolic form below, but as they aren't mere negations of each other, it was a little tricky. Is there any mistake below (especially for 'isolated ...
1
vote
1answer
134 views

What is the definition of a “Circular Wedge”?

In Ahlfors' Complex Analysis, chapter 3, section 4, the author claims that a region whose boundary consists of two circular arcs with common end points is either a "circular wedge" or its complement, ...
1
vote
1answer
46 views

Holomorphic extension to a closed half-space

While reading D. Zagier's expository paper on the proof of the prime number theorem given by Newman, I encountered the following terminology problem : let $f$ be a holomorphic function defined in an ...
1
vote
1answer
101 views

Definition of Holomorphic map of complex manifolds

Let $X,Y$ be complex manifolds and let $f: X \rightarrow Y$ be a continuous map. When exactly do we say that $f$ is "holomorphic"? I am interested in the basic definition (possibly using charts), not ...
4
votes
0answers
67 views

What is the term used for space of analytic functions?

I deal with analytic functions in the unit disc represented as the series $\sum_{n=0}^\infty u_n z^n$, where the coefficients $u_n$ satisfy the condition $\sum_{n=0}^\infty n^\alpha|u_n| < \infty$ ...
0
votes
2answers
78 views

What is a conformal mapping with $\infty$ in its image?

Suppose we have a bijection $f$ between two open sets in $\mathbb{C}\cup\{\infty\}$, for example $U=\{|z|<1\}$ and $\Omega$, with $\infty\in\Omega$. Let $f(0)=\infty$. What do we mean by saying ...
0
votes
1answer
83 views

“fluent” functions

In an old mathematics book (Ritt, 1948, p.5) I have come across the notion of "monogenic analytic" and "fluent" functions. These are complex valued functions. Has anyone heard of these terms before? ...
3
votes
1answer
397 views

What does it mean for a function to be bounded near $\infty$?

Suppose $f(z)$ is some analytic function which is bounded near $0$. Then $f(1/z)$ is bounded near $\infty$. What exactly does that last statement mean practically? Does it mean $|f(1/z)|$ is bounded ...
3
votes
0answers
201 views

Etymology of the word “pole”?

In his book Control System Design, Bernard Friedland writes (section 4.2, page 115): The roots of the denominator [of a rational function] are called the poles of the transfer function because ...
1
vote
0answers
235 views

Monodromy theorem in Princeton lectures in analysis

I am very curious to know if what is called the "Monodromy Theorem" or "Riemann Monodromy Theorem" is also known by some other name. For all my complex analysis requirements I have always fallen ...
6
votes
1answer
177 views

Nomenclature in complex analysis

I am having a little confusion on the naming of functions in complex analysis. If $f$ is a holomorphic function on the complex plane and its domain is the complex plane then it's called an entire ...
3
votes
3answers
741 views

What is “entire finite complex plane"?

The question is from the following problem: If $f(z)$ is an analytic function that maps the entire finite complex plane into the real axis, then the imaginary axis must be mapped onto A. the ...
77
votes
5answers
6k views

What is the Riemann-Zeta function?

In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?