# Tagged Questions

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

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 ?
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### 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 ...
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### $\prod_{n}f_{n}$ 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 ...
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### $2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$

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 ...
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### 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 ?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### “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 $f(x)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...