Tagged Questions

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1answer
41 views

Is there a text book containing a self-contained and complete proof of the Jordan Curve theorem?

I seem to remember (in my undergraduate years) encountering a book on complex analysis which contained a proof of the Jordan Curve Theorem, building up from first principles - so self-contained and ...
3
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1answer
38 views

Dual of holomorphic functions (with the $L^1$ topology)

Let $\Omega$ be a connected domain of the complex plane, and let $E$ be the vector space of integrable holomorphic functions on $\Omega$. Then it can be checked that $E$ is a closed subspace of ...
5
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3answers
76 views

A boundary version of Cauchy's theorem

I am looking for a reference for the following theorem (or something like it) that is not Kodaira's book. Let $D$ be a domain and $\overline{D}$ be it's closure. Suppose that $f:\overline{D} ...
0
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1answer
57 views

What are the prerequisite for understanding complex analysis?

Which should I complete first before complex analysis? I am following Visual Complex Analysis by Tristan Needham. Is there any easier book?
1
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1answer
33 views

Continuous dependence of zeros on a parameter

Let $F:I\times J\to\mathbb{R}$ be a $C^k$ (or analytic) function, with $I,J$ real open intervals. Set $f_\lambda(x):=F(\lambda,x)$ and consider the parametric equation $$f_\lambda(x)=0\,.$$ Assume its ...
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0answers
32 views

theorems on analytic extension

I am a probability student and I have forgotten much of complex analysis I have learnt when I was an undergraduate. I have recently seen integrals evaluated using analytic extensions techniques when ...
0
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2answers
64 views

Whats the connection between functions with curl 0 and holomorphic functions

When I first saw the Cauchy-Riemann differential equations they remind me on the conditions for the curl of a function to be zero. Here some notation I will use: $$\frac{\partial f}{\partial x} = ...
0
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0answers
26 views

Locating all zeros of a complex analytic function in a bounded domain

There are some papers on this topic, e.g., [Davies1986] Davies, B. Locating the zeros of an analytic function Journal of Computational Physics, 1986, 66, 36 - 49. [Gillan2006] Gillan, C.; ...
3
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1answer
76 views

Moduli Spaces of Higher Dimensional Complex Tori

I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action. Similarly, I have ...
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0answers
75 views

A difficult, concise, and applicable complex analysis book.

I am currently going through Spivak's Calculus on Manifolds. I love the concision (only around 150 pages), and the problems are at just the right level for me (although I'd still be very happy if they ...
4
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1answer
45 views

Literature request - Classification of periodic holomorphic functions

For a seminar, I received the assignment to present the classification of periodic holomorphic/meromorphic functions. I have access to a limited amout of resources that I receive from my lecturer - ...
5
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1answer
115 views

Book Searching in Complex Analysis

I'm searching for a problem book in complex analysis published by MIR. It was recommended by my professor (when I asked for a Demidovich equivalent in the field), but he did not remember the exact ...
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0answers
34 views

A text or book in holomorphic foliations and vector fields over complex manifolds

For my master's degree dissertation, I am going to study some implications of the paper "SOME REMARKS ON INDICES OF HOLOMORPHIC VECTOR FIELDS" written by Marco Brunella. I just started it and I'm ...
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4answers
195 views

Necessarily complex analytic proofs in algebra.

Does anyone know of an example where complex analysis is necessary to prove something in algebra? I would be particularly interested in results from group theory or Galois theory. In an ideal ...
11
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1answer
289 views

Images of compact subsets in the plane

Let $K$ be an infinite compact subset of $\mathbb{C}$. Is it true that there exists a sequence $(f_n)_{n>0}$ of functions holomorphic in some neighborhood of $K$, such that the images $f_n(K)$ are ...
1
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1answer
45 views

Suggestion for references

Can anyone suggest me some good references for studying HENON MAPS. I have no information about these maps at all. So please suggest some references for going through its basics.
5
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1answer
118 views

Hausdorff Dimension of Arbitrary Julia Set

I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$. I know this question is known for a number of ...
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0answers
20 views

About continuity of fixed points under parametric changes

The motivation to this quetion can be found in: Rotation and fixed points (2) My question is: I have trouble understanding and elaborate the last remark in the answer: ...about continuity of fixed ...
5
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1answer
83 views

How does one see the topology of a Riemann surface from the graph (assuming one can picture $\mathbb R^4$)?

Given a function $f:\mathbb C\to\mathbb C$ which we will assume is analytic, we have an embedding $f\subseteq\mathbb C\times\mathbb C\cong\mathbb R^4$ of a surface. My question is with regards to how ...
0
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1answer
89 views

Harmonic Extension

Let be $u$ a harmonic function defined on an open set $\Omega \setminus \{p\} \subset \mathbb{C}$ of the complex plane. Show that if $u$ is bounded in a neighborhood of $p$ then $u$ admits a harmonic ...
104
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3answers
5k views

The Integral that Stumped Feynman?

In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
7
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1answer
169 views

Books on complex analysis

Is there any book on $1$-dimensional complex analysis, where all is written in the language of sheaf theory? It's clear, that a lot of constructions can be formulated in simplier way using it. There ...
6
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1answer
134 views

Complex Analysis textbook - specific criteria

This is my first question on this site and I hope I won't screw it up. I'm looking for a text (textbook, lecture notes etc.) on Complex Analysis that meets some very specific desiderata. I've already ...
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0answers
55 views

$\sum_p z^p$ where $p$ is prime

I've started reading Shakarchi's Complex Analysis, and I thought about something interesting. If I haven't mistaken, for any subsequence $A\subset \mathbb{Z}^+$, $\sum_{n\in A} z^n$ has radius of ...
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1answer
103 views

How should I study complex (ordinary) differential equations (reference-request related)?

I'm currently studying complex analysis at a university (undergraduate) but my lecturer is doing a little bit of ordinary differential equation theory too, inclduing hypergeometric functions. As I ...
2
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1answer
91 views

Poisson integral on $\mathbb{H}$ for boundary data which is orientation-preserving homeomorphism of $\mathbb{R}$

Let $f$ be a real-valued function (in my case, an orientation-preserving homeomorphims of $\mathbb{R}$) on the real line $\mathbb{R}$ which is not in any $L^p$ -space. Let us take the simplest example ...
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2answers
250 views

Complex analysis book with a view toward Riemann surfaces?

I am considering complex analysis as my next area of study. There are already a few threads asking about complex analysis texts (see Complex Analysis Book and What is a good complex analysis ...
2
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5answers
404 views

Does $i^i$ and $i^{1\over e}$ have more than one root in $[0, 2 \pi]$

How to find all roots if power contains imaginary or irrational power of complex number? How do I find all roots of the following complex numbers? $$(1 + i)^i, (1 + i)^e, (1 + i)^{ i\over e}$$ EDIT:: ...
2
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3answers
309 views

how to show that power series is analytic inside the radius of convergence?

Let $f(z) = \sum a_n z^n$ be a power series with radius of convergence $R$. How do we show that $f$ is analytic in the circular region of radius $R$?
2
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1answer
56 views

An entire function as the projection in a fibre bundle

Let $f$ be a non-constant entire function. I'd like to view $f$ as the projection mapping in a fiber bundle, where the base space is $X =f(\mathbb{C})$ (which according to Picard is either ...
0
votes
1answer
75 views

global irreducible decomposition of an analytic set

Let $M$ be a complex manifold (or a complex analityc space) and $Z$ be an analytic subset of $M$. By Noetherianity of the rings of germs of analytic functions at a point we know that $Z$ has finitely ...
0
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1answer
160 views

A consequence of Runge's theorem

I'd like to have a reference for the proof of the following fact of complex analysis. I think it follows from Runge's theorem, but I don't know how to prove it. Fact. Let $U \subseteq V \subseteq ...
9
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7answers
1k views

Complex Analysis Book

I want a really good book of Complex Analysis, for good understanding theory. There are many complex variable books that are only list of identities and integrals, I hate it. For example Munkres is a ...
11
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2answers
495 views

Infinite products - reference needed!

I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to ...
0
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2answers
215 views

Why (finite) Blaschke products are actually rational fractions?

I have found in several books the following affirmation : Let $f: \Delta \rightarrow \Delta$ be a non constant holomorphic function that extends continuously to $\overline{\Delta}$, $\Delta$ being ...
6
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1answer
246 views

When can we find holomorphic bijections between annuli?

I'm self-studying some complex analysis, and apparently holomorphic bijections between two annuli exist precisely when the ratios of the radii are the same. More exactly, if ...
0
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0answers
80 views

Square root of biholomorphic mappings between multiply connected domains

I've been looking in the literature for a reference on the following, but without success : Let $\Omega_1$ be a bounded finitely connected domain in the complex plane. Suppose that the boundary of ...
2
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0answers
148 views

Cauchy's integral formula for operators

I study this article : A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model. Massimo Campanino and Abel Klein. Comm. Math. Phys. 104 ...
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1answer
166 views

The Implicit Function Theorem for complex polynomials

I'm looking for a reference that proves implicit function theorem for polynomials in two variables over the complex numbers via the real version. Such a theorem is needed, for example, in the theory ...
7
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3answers
228 views

Sources on Several Complex Variables

I have searched the past entries about sources on SCV but couldn't find about this topic. If I am not careful enough, sorry for this! We are using Hörmander's book which is really hard to follow. ...
1
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1answer
134 views

Suggestions for a Global Analysis book

can somebody tell me some good books or lecture notes in "global analysis" ? I am a newcomer in this subject. thanks in advance. greetings trito
1
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1answer
43 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
2
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1answer
78 views

Asymptotic study of complex integrals

Anyone know a good reference for the asymptotic study of integrals of the form $\int_{\Gamma}f(s)e^{ng(s)}ds$ , $n\to\infty$, where $f(z)$ and $g(z)$ are analytic functions in the domain containing ...
9
votes
1answer
191 views

Justifying taking the limit of a keyhole contour

In Stein's Complex Analysis book, he proves Cauchy's integral formula (and a few others) by use of a keyhole contour argument (see page 46). He does not explain why the integral of the limit path is ...
17
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5answers
1k views

Picard's Little Theorem Proofs

Picard's little theorem says that If there exist two complex numbers $a,b$ such that $f: \Bbb{C} \to \Bbb{C}\setminus \{a,b\}$ is holomorphic then $f$ is constant. I am interested in proofs for ...
4
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0answers
236 views

Organizing types of functions by their calculus-related properties, in diagram form?

Does anyone know of a diagram that displays and organizes categories of functions according to their calculus-related properties (e.g. continuous, $C^\infty$, degrees of differentiability and ...
3
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0answers
132 views

Change of variables in line integral with abs. value

Let $\gamma : I \rightarrow \mathbb C$ be a path. Let $g: \mathbb C \rightarrow \mathbb C$ be a biholomorphic map. Let $f$ be a holomorphic function. Consider the integral $$ \int_{g\circ \gamma} ...
1
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1answer
203 views

Roots of truncated taylor series of exp and lambertW

If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get very close approximations to the roots of the scaled truncated taylor series of $\exp$. Here W is the lambertW function, $e$ ...
3
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3answers
507 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 ...
10
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1answer
387 views

Radius of convergence of power series

Given a meromorphic function on $\mathbb{C}$, is the radius of convergence in a regular point exactly the distance to the closest pole? As Robert Israel points out in his answer, that this is of ...

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