# Questions tagged [cv.complex-variables]

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

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50 views

### Metric with singularities on Riemann Surfaces and the associated Laplacians

I have asked this question on Math Stack Exchange
Metric with singularities and associated Laplacian
but I have not got any answers/comments, therefore I post this question on the MO.
Suppose $M$ ...

**0**

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**0**answers

28 views

### Is there a way to categorise the valleys of a holomorphic function (potentially of $\geq 2$ variables) (multidimensional steepest descent)

More specifically, I am particularly interested in the question: given some $f:\mathbb{C}^n \to \mathbb{C}$, can we categorise $\mathbb{C}^n$ by which valley the steepest descent curves of a point (...

**5**

votes

**1**answer

183 views

### Dependence of a solution of a linear ODE on parameter

Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...

**2**

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**0**answers

106 views

### Is there any neat way to calculate the Fourier transform for an inverse Vandermonde determinant?

$$\mathrm{PV}\int_{\mathbb R^n} \frac{e^{-i\langle w, x\rangle}}{\prod_{j<k}(x_k-x_j)}dx=?$$
Other than integrate this term by term (which might look crazy)?
Let $f(x)=1/\prod_{j<k}(x_k-x_j)$, ...

**3**

votes

**2**answers

356 views

### Reference request: Oldest complex analysis books with (unsolved) exercises?

Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ...

**1**

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**1**answer

113 views

### An equation with Gamma Euler function in critical strip

Let
$$
D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \}
$$
that is the critical strip without critical line.
I have to find if the following equation, with ...

**3**

votes

**0**answers

103 views

### Injective resolution of the ring of entire functions

Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis.
I would think that the injective dimension ...

**3**

votes

**1**answer

48 views

### Rational approximation on rotation invariant compact subsets of complex plane

What does the Vitushkin's theorem say about the equality $A(K) = R(K)$ in the special case when $K$ is rotation invariant? More precisely, what are necessary and/or sufficient conditions on $\{|k|: k \...

**2**

votes

**2**answers

101 views

### Coefficients of entire functions with specified zero set

Let $Z \subseteq \mathbb{C}$ without limit point. By the Weierstrass factorization theorem there is an entire function $h$ those zero set is $Z$. Let $a_n > 0$ be a sequence where $\lim_n \sqrt[n]{...

**3**

votes

**1**answer

156 views

### Local phase statistics of the nontrivial Riemann zeros

(The question is inspired by Owen Maresh's post)
The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$.
Numerical results on the first 10000 zeros suggest ...

**1**

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**0**answers

100 views

### Defining integrals by residue theorem

I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....

**0**

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141 views

### An (unusual) uniqueness theorem for analytic functions

Let $L$ be the class of analytic functions in $\mathbf{C^*}$
with positive Laurent
coefficients:
$$f(z)=\sum_{-\infty}^\infty c_nz^n,\quad 0<|z|<\infty,\quad c_n\geq 0.$$
Each $f\in L$ has a ...

**0**

votes

**1**answer

74 views

### Solving or bounding the real part of the integral $\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$

I would be interested in finding a closed form or, at least, bounding (in terms of $m$ as it becomes larger) the real part of the following itnegral:
$$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} ...

**3**

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35 views

### Modulus of image of a curve family in a rectangle

I don't expect to get a positive answer to this question but I may as well try.
Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...

**0**

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**1**answer

66 views

### meromorphic extension of dirichlet series

Suppose $\{a(n)\}_{n\ge 1}$ is a bounded complex sequence. Let $\phi(s)=\sum_{n\ge 1} \frac{a(n)}{n^s}$. Obviously, the Dirichlet series $\phi(s)$ is absolutely convergent for $\mathcal{R}(s)>1$. I ...