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        Questions tagged [cv.complex-variables]

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

        3
        votes
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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 $M$ ...
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        votes
        0answers
        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 (...
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        votes
        1answer
        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 (...
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        votes
        0answers
        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 $f(x)=1/\prod_{j<k}(x_k-x_j)$, ...
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        votes
        2answers
        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 ...
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        1answer
        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 ...
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        votes
        0answers
        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
        1answer
        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 \...
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        votes
        2answers
        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
        1answer
        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 ...
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        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....
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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 ...
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        votes
        1answer
        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} ...
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        votes
        0answers
        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 $l,h>0$. ...
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        votes
        1answer
        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 ...

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