# Tagged Questions

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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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 - ...