# Questions tagged [cv.complex-variables]

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

1,985 questions
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### Do analytic functionals form a cosheaf?

Let $X$ be a complex-analytic manifold. Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
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### What is the Laurent series of z+(1/z)? [on hold]

What is the Laurent series of z+(1/z)? Is it just the series itself?
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### When is entire function bounded on a ray?

Let $f(z)=\sum_{n=1}^\infty c_n z^n$ be entire function on the complex plane. May we express the property $\int_0^\infty |f(x)|^2 /x dx<\infty$ or some other property controlling the behavior for ...
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### On the roots of Bernoulli polynomials

Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It ...
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### Compatible solution of PDE

Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be ...
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Let $\sum_{j=1}^{\infty}a_{j}$ be a convergent series of positive numbers and $\{z_{j}\}_{j=1}^\infty$ a closed discrete subset of the open unit disc $\mathbb{D}$. Then $h(z):=\sum_{j=1}^{\infty}\frac{... 0answers 130 views ### Chern number of projection-Topological magic in physics I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is$-1$) and the full paper can ... 0answers 45 views ### Entire analytic functions with entire analytic Fourier transform, and corresponding distributions I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for$\delta$-distributions supported at complex ... 2answers 145 views ### Density of Lacunary Functions I'm curious whether lacunary functions (functions in the complex plane that are holomorphic in some open ball about the origin, but cannot be analytically extended past that ball) are typical or the ... 1answer 68 views ### Bringing a Heun equation into canonical form It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly$4$regular singular points may be suitably ... 3answers 199 views ### Origin of term Ahlfors-David regular Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let$q>0$. A metric space$(X,d)$is Ahlfors(-David)$q$-... 1answer 131 views ### Compilation of representations of holomorphic functions Holomorphic functions are my muse. As my muse, I love drawing them different ways. Allow me to frame this as though an artist talking about his muse. A holomorphic function$f$on the unit disk$\...
Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole ...