0
votes
1answer
18 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
3
votes
1answer
37 views

Book Recommendations for Picard Big and Little Theorems

Does anybody have book recommendations for reading about Picard's Little and Big Theorems? Preferably, I am looking for a book that is intended for an undergraduate/first year graduate student who ...
0
votes
2answers
22 views

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients?

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients? Where can I read a proof? p.s. I don't even see the answer to the first question with a google ...
2
votes
1answer
34 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
4
votes
0answers
48 views

Does anyone have a good reference on calculating contour integrals around the unit circle (numerically or otherwise)?

I am looking for a reference that will help me calculate contour integrals around the unit circle or other curve. I have a particularly ugly function which isn't likely to have a nice closed form so I ...
2
votes
3answers
56 views

Recommendations for books on complex analysis and on measure theory?

I'm looking for a book on complex analysis that has a similar writing style to either Terry Tao's Analysis II or Nathan Jacobson's Basic Algebra series. I have found both of these extremely easy to ...
0
votes
0answers
19 views

Banach-valued holomorphic functions [duplicate]

Let $X$ be a Banach space. Can we define holomorphic functions $f:\mathbb{C}\to X$ by the notion of derivability i.e. $$\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h}$$ Do we still have equivalence between ...
0
votes
2answers
71 views

Basic question on complex integration

I have a very basic question on complex integration. How is the definite integral $$ \int_{z_1}^{z_2}{f(z)dz} $$ $z \in \Bbb{C}$ to be interpreted in the absence of a specific path over which ...
1
vote
1answer
15 views

Extending a holomorphic function to a radial limit function for almost every angle

I've read in several places about the "well known theorem" which states that a holomorphic function on the (open) unit disk $D=\{z\in\mathbb{C}:\ |z|< 1\}$ can be extended to its boundary on almost ...
0
votes
0answers
46 views

Texts for Complex Analysis

I am interested in reading about complex dynamics, Riemann surfaces, and related subjects, but I lack complex analysis as a prerequisite. I want a text that is rigorous and challenging (e.g. not a ...
4
votes
4answers
260 views

Best complex analysis references?

I own Gamelin's 'Complex Analysis', but I'm having a bit of a hard time understanding it. I have also tried watching MIT Open Courseware videos on the subject, but I easily get lost. Are there any ...
2
votes
0answers
33 views

Fractional linear transformations and the extended complex plane in a more abstract context?

Does anyone know of an "abstract algebra-esque" treatment of the extended complex plane and the Mobius transformations? I am studying complex analysis now, and I am a little frustrated that my ...
2
votes
1answer
57 views

Recommend textbooks that expain branch cut, Riemann surface and contour integration with branch cut in detail

I read several textbook on complex analysis, but few of them explain the branch cut and Riemann surface in detail and treat the contour integration with branch cut. But this is very important for many ...
4
votes
4answers
137 views

A question regarding Frobenious method in ODE

Suppose $b(x),c(x)$ are real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
3
votes
0answers
89 views

Which book is better, Rudin or Ahlfors? [closed]

As a beginner, with no history in analysis, what book is better for self teaching; Rudin or Ahlfors? Thanks!
3
votes
0answers
61 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
0
votes
1answer
51 views

Find a conformal map from the exterior of the closed unit disk to the unit disk

Question: Find a conformal map from the exterior of the closed unit disk to the unit disk. Also, prove that it is indeed a conformal map (bijective and holomorphic along with its inverse). I missed ...
1
vote
0answers
39 views

Historical context: The Fresnel integrals

The evaluation of the Fresnel integrals has been done a plethora of times both on this site, and numerous other places. The two main ways of evalutating these integrals has either been with some ...
1
vote
0answers
16 views

doubt about definition of holomorphic polynomials

In a topic of several complex variable theory (in particular functions on $\mathbb{C}^2$), I came across a term homogeneous holomorphoic polynomial. By the word, I think it is a polynomial in complex ...
0
votes
0answers
38 views

Is there a good introductory complex-analysis text in general setting, namely Riemann sphere?

I have studied first 1~3 chapters of some complex analysis texts (Ahlfors, Conway, Silverman) Well, i specially like Ahlfors in many ways but this text doesn't seem to develop a theory in a general ...
0
votes
2answers
47 views

Riemann Sphere/Surfaces Pre-Requisites

I have recently developed a large interest in everything to do with Riemann Sphere/Surfaces. I wish to understand the topic quite well but I know that I will need to read a good number of books on ...
1
vote
0answers
61 views

Cramer's Rule with complex system of equations

Given a 2x2 system of complex equations with one unknown, $z$, written as a 2x2 matrix, $A$, would the system have infinitely many solutions iff $\det(A_x)=\det(A_y)=\det(A)=0$? Or is there more to ...
2
votes
1answer
36 views

I am interested on the functions $g:ℂ×ℂ→ℂ$ of the form $g(x+iy,x-iy)=g(z,\overline{z})$

I am interested on the functions $g:ℂ×ℂ→ℂ$ of the form $$g(x+iy,x-iy)=g(z,\overline{z})$$ My question is about requesting some references dealing with this type of functions.
2
votes
1answer
56 views

Removable singularities for continuous functions

Let $f: D - K \rightarrow \mathbb{C}$ be holomorphic, where $D$ is a planar domain and $K$ is a compact subset of $D$. Suppose that $f$ extends continuously to all of $D$. On which conditions on $K$ ...
1
vote
5answers
198 views

What books on analysis after someone has finished all 3 by Rudin?

What books on analysis would people recommend after someone has finished all three by Rudin (Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis)? I am looking for ...
2
votes
1answer
52 views

$\sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0$

Let $f \in H(\mathbb{C}- \{ z_{1}, \dots, z_{n} \})$. I need a proof of the fact that $$\sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0.$$ Where can I find it ?
2
votes
0answers
30 views

Accurate computation of arcsec near branch points

The direct numerical implementations of the usual definitions of the complex $\mathrm{arcsec}(z)=\arccos(1/z)$ and similar for $\mathrm{arccsc}(z), \mathrm{arcsech}(z), $ etc are not accurate near ...
1
vote
0answers
58 views

$\prod_{n}f_{n}$ converges uniformly $\Rightarrow $ $\sum_{n}\mathrm{Log}(f_{n})$ converges uniformly

Let $\prod_{n}f_{n}$ be an infinite product of holomorphic functions on a given domain $\Omega$ converging uniformly on compact subsets of $\Omega$ to $f$. Then is it true that ...
1
vote
1answer
55 views

$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$

Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$ then $f$ is linear. I know this is a well-known ...
3
votes
2answers
49 views

Relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$ [duplicate]

What is the relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$ where $a_{n} \in \mathbb{C} \ \forall n$ ? Where can I find some references about this topic ?
0
votes
1answer
81 views

Prove the Jordan lemma i.e. $\int e^{-R\sin{\theta}}< \pi/R$

In complex variables my instructor wrote on the board "Jordan's Lemma", and then, somewhat imprecisely, $$\int e^{-R\sin{\theta}}< \pi/R \;\;\;\; \text{ e.g. } \int \frac{s \sin{x}}{x^2 + 2x + ...
4
votes
1answer
60 views

How to choose contour in $\mathbb{C}$ to do Residue Integration.

I'm almost sure that there's not any simple way to answer this question, but I'll try. I'm studying complex variables and the method of calculating improper integrals with residues but I'm struggling ...
3
votes
2answers
172 views

A gentle textbook of complex analysis

Is there a gentle textbook of complex analysis ? Something equivalent to Larson's Calculus (or Stewart's). I have Schaum's Outline of Complex Variables (Spiegel-Lipschutz), and it's not bad.
2
votes
2answers
43 views

Stability of Analytic Continuation

Let $f(z)$ be an analytic function in an open set $U\subset\Bbb{C}$. Recall that an analytic continuation of $f$ is a pair $(F,V)$ such that $U\subset V\subset\Bbb{C}$, $F$ is analytic on $V$, and ...
2
votes
1answer
127 views

Complex Analysis and Probability Theory

My question is a general one. I know that in complex analysis we find some very powerful theorems but given that my main area of study is Statistics and Probability, does complex analysis have ...
1
vote
3answers
237 views

Roadway and book recommendations to math study.

I had some calculus, linear algebra and complex analysis courses back in college. But it is not comprehensive. And I felt that my college math was not taught in a logical sequence (maybe because my ...
4
votes
1answer
44 views

Boundary data of the modulus of a holomorphic function

Let $f$ be a non-vanishing holomorphic on the unit disk $D$. Suppose $|f|$ converges to a measure $\mu$ on $\partial D$ as $|z|\rightarrow 1$, in the sense that $$ \int_{\partial D} |f(r z)| \phi(z) ...
1
vote
1answer
89 views

Book suggestion- complex analysis -conformal mapping.

I am studying complex analysis. And I am using J. Bak and D.J. Newman's book.(springer) And now my studying topic is conformal map. In addition to this book, I want to learn other book names which ...
1
vote
0answers
107 views

Monodromy Groups of Differential Equations

I have heard that monodromy groups and analytic continuation can be used to construct new solutions to a differential equation from a particular solution. What references (textbook, or papers) could I ...
3
votes
0answers
76 views

Explicitly realizing Riemann surfaces as a quotient of the upper-half plane

Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic ...
1
vote
3answers
103 views

Complex book suggestions

I take complex analysis course. And my instructor use -Bak and Newman's complex analysis book, springer. This book explains too fast and superficially. Please give me book suggestions which are the ...
3
votes
1answer
353 views

Cauchy-Goursat theorem, proof without using vector calculus.

On the wikipedia page for the Cauchy-Goursat theorem it says: If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proved as a ...
2
votes
1answer
51 views

Riemann surface intuition.

In my complex variables notes it says that the multivalued $n$-th root function $w=z^{\frac{1}{n}}$ becomes single-valued on an appropriately constructed Riemann surface. It says how to go about ...
2
votes
0answers
139 views

“Analytic continuation” of function of two real variables

Consider the two situations (A) If I have a real analytic function $f(x)$ of real $x$, the series $$\sum_{n=0}^{\infty}\frac{(z-x_0)^n}{n!}f^{(n)}(x_0)$$ gives an analytic continuation of $f(x)$ ...
2
votes
0answers
206 views

Cauchy's theorem for integral homotopic closed curve in $G\subset\mathbb{C}^n$.

Recall Cauchy's theorem (third version in the Conway's book "Function of one complex variable", thm 6.7. page 90 in the second edition): Let $f$ be an analytic function on $F\subset\mathbb{C}$ and ...
4
votes
2answers
441 views

Multivariate Residue Theorem?

Is there an extension of the residue theorem to multivariate complex functions? Say you have a function of $n$ complex variables $s_{n}$ and you wish to integrate it over some region in ...
2
votes
1answer
57 views

Book searching in Pluripotential theory

Can anyone recommend me a book on pluripotential theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I want to understand why ...
0
votes
1answer
32 views

About a reference collecting the main properties of the modulus and the argument of $f$

Let $f$ be an analytic function in the whole complex plane. We can write $f$ in its polar form: $$f(z)=ρ(z)exp(iθ(z))$$ My question is about a reference collecting the main properties of the modulus ...
3
votes
5answers
312 views

Rigorous Textbook for Introduction to Complex Numbers/Analysis?

Does anybody know where I can find a rigorous textbook on developing complex numbers/analysis? I'm currently working through Needham's Visual Complex Analysis, which is interesting but non-rigorous. ...
0
votes
2answers
244 views

Schwarz's lemma $\Rightarrow$ an analytic conformal map UHP$\to$UHP must be an FLT?

I read a solution to a conformal mapping problem that made the claim, "Schwarz's lemma implies that any analytic conformal map taking the upper half-plane to the upper half-plane must be a fractional ...