# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

1,979 questions
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### Residues of Zeta-like Function

I'm looking for the residues of the following function $$s \mapsto\sum^\infty_{m,n =1} (m+n) \left[ amn + (m-n)^2 \right]^{-s}$$ at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real positive ...
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### Complex differential equations

I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs. Mostly, I'm just ...
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### Derivative of complex matrix pseudo inverse with respect to real and imaginary components

I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$. I am interested in evaluating the ...
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### Non-compact analogue of Hartog's extension theorem?

Suppose a function $f(z,w)$ is analytic in the open polydisk $\Delta^2$ with $\Delta = \{z \in \mathbb C | |z| < 1 \}$. I am interested in the limit $f(z,w)$ as $w \to 1$. This limit may be ...
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### Induction principle on proving an inequality

If $P(z)$ having no zeros in $|z|<1,$ then $$\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}\leq \frac{n}{2}.$$ Can we prove this by induction on $n$? or is there any alternative way? Attempt at ...
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### The generalization of Hartogs' Theorem

I know a version of Hartogs' Theorem in the book An introduction to complex analysis(p 30) by Hormander, namely Hartogs' Theorem when K compact with complement being simply connected I also have ...
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### compare N(f,a,r) with T(f,r)

I'm reading William Cherry and Zhuan Ye's book 'Nevanlinna's theory of value distribution, the second main theorem and its error terms'. In Section 1.12, they explains why $N$ and $T$ is used in ...
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### Vector valued disc “algebra”

I am interested in a vector-valued form of the disc "algebra" (which in this setting is not in general an algebra, hence the scare quotes). Let $E$ be a Banach space, and let $A(\mathbb D,E)$ be the ...