# 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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### Conditions for an improper integral of a real-analytic function to be real-analytic

Let $U$ be a complex domain and suppose $f:U \times \mathbb{R} \rightarrow \mathbb{R}$ is continuous and, for all $t$, is real-analytic in $s$. That is, $f(s, t)$ is continuous and $g_t(s):=f(s, t)$ ...
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### Existence of analytic function in disk algebra

Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?
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### Generalization of residue theorem to arbitrary complex manifolds

If $f : \mathbb{C}\rightarrow \mathbb{C}$ is an everywhere complex-analytic function, then along a closed contour $C$, $\int_{C} f(z)dz = 0$. Is there a generalization of this theorem to arbitrary ...
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Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$. How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{... 0 votes 0 answers 77 views ### Difference of a curve with itself [closed] I asked this question on stack exchange but did not get any response.To re iterate,Consider a closed ,smooth curve$\gamma(t),a\leq t \leq b.$For any$t \in [a,b]$,let us define :$$z(t)=x(t)+iy(t)$$ ... • 612 18 votes 2 answers 958 views ### Laurent series in several complex variables Is there a good generalisation of Laurent series for several complex variables? I am interested in generalised power series that have some terms with negative powers, but not too many. In single ... • 587 2 votes 1 answer 126 views ### Reference for asymptotic estimates In the way of studying an enumerative problem I have found that I have to estimate the Taylor coefficients of functions of the following form. For two polynomials$P(x)$and$Q(x)$with$P(0)=Q(0)=1$, ... • 1,296 0 votes 0 answers 105 views ### How to prove an equality involving Laguerre polynomials Assume$\mu<0$. Let$L^\alpha_n(x)$be Laguerre polynomials of type$n$and$f\in L^2(\Bbb C)$. How to prove that $$\sum^\infty_{k=0}\int_{\Bbb C} f(z)\frac{1}{2(2k+1)-\mu}L^0_k(\lvert z\rvert^2)... 3 votes 0 answers 97 views ### Converse theorem for zeta universality Voronin's Universality Theorem for \zeta(s) is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ... • 131 1 vote 1 answer 101 views ### Bott-Chern cohomology for singular complex spaces I'm reading the book 'An Introduction to the Kahler-Ricci Flow' (Lecture Notes in Mathematics 2086). They discuss Bott-Chern cohomology on complex spaces: Let X be a complex space(i.e. analytic ... • 263 0 votes 0 answers 32 views ### For fixed B > 0, asymptotically describe the upper half plane complex zeros of B\cos(\omega z) = iz as \omega \to \infty Long story short, I am seeking to evaluate an oscillatory integral via complex methods, whose partial fraction expansion has B\cos(\omega z) - iz in the denominator, where B, \omega are positive ... • 143 0 votes 0 answers 162 views ### On \sum_{\rho\in D} \text{dist}(\rho)=\frac{1}{2\pi i} \int_{\partial{D}}\log \zeta(s)\ ds Let D denote a closed two dimensional figure as: D=2+iT\to 2\to 2-\delta\to 2-\delta+i(T-\delta)\to \frac{1}{2}+\epsilon+i(T-\delta)\to\frac{1}{2}+\epsilon\to\frac{1}{2}-\epsilon\to \frac{1}{2}-\... • 11 0 votes 0 answers 94 views ### Criterion to decide whether a function is algebraic For Christol's theorem see 1, if a power series is algebraic over every fields of characteristics p, is it algebraic over fields of 0, by Robinson's principle? Update: The power series is in \... • 2,670 1 vote 0 answers 25 views ### Application of the \operatorname{BMO}, H^1 duality Let f\in \operatorname{BMO}(\partial \Delta), then there exists a Carleson measure \mu in \Delta such that$$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\... • 111 3 votes 1 answer 71 views ### Non-holomorphicity of Hecke regularization of weight-2 level-1 Eisenstein series Originally asked on stackexchange since I figured it was a very elementary and silly error, but since I haven't had any response or interest in a couple days, I thought I'd post it here. I have a &... • 1,550 3 votes 0 answers 127 views ### How to correctly renormalize this function at the pole$x=1$? Evaluating:$\sum_{n=1}^{\infty} e^{e^n}$So I was considering the divergent everywhere but 0 power series $$f(x) = \sum_{n=0}^{\infty} e^{e^n} x^n$$ Now one can do the following "questionable" manipulation $$f(x) = \sum_{n=0}^{\... 0 votes 0 answers 57 views ### To find a DFT for complex functions on a semigroup For a certain commutative semigroup of integer size n, G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y)), consider all complex functions on it, denoted by \mathbb C[G] or \mathbb CG. ... 19 votes 1 answer 616 views ### On the equation \zeta(s) = F(s)+F(s+1) Define the function F(s) as the Dirichlet series$$ F(s) = \sum_{n=1}^\infty \frac{1}{(n+1)n^{s-1}}, $$which converges for \operatorname{Re}(s)>1. Has anyone seen/studied this function before? ... • 2,166 3 votes 2 answers 117 views ### Is every planar bounded C^2 domain finitely connected? Let \Omega \subset \mathbb R^2 be a bounded C^2 domain. Is \Omega then finitely connected? As I learned recently a domain in \mathbb R^2 is finitely connected iff “[its] complement has ... • 243 2 votes 0 answers 50 views ### Analytic continuation of Fredholm integral equations I am considering the equation of Fredholm type$$ f(x) = \lambda \int_\zeta \prod(y-l_i(x))^{\alpha_i} f(y) dy\label{1}\tag{1} $$where \zeta is an element of 1st homology of the complement to ... • 31 1 vote 0 answers 39 views ### Holomorphic "quasi-interpolation" of a function sequence I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let V \subset \mathbb{C} be a complex neighborhood of [0,1]. Assume there is some bounded ... 3 votes 1 answer 233 views ### Relationship between two kinds of classifications of Riemann surfaces There are two kinds of classifications of Riemann surfaces. Classification 1: Let M be a Riemann surface. We will call M: elliptic iff M is compact (= closed); parabolic iff M is not compact ... • 39 11 votes 3 answers 654 views ### Can computers find zeros of order 2? We assume we are given an entire function f: \mathbb C \to \mathbb C with f(0)=1 and f'(0)=0 and f is real on the real axis. We assume (as a fact about f, that we want to demonstrate ... • 614 7 votes 1 answer 187 views ### Real analytic subvariety in complex manifold which is complex outside of its singular set Let M be a complex manifold, and Z \subset M a closed real analytic subvariety. Suppose that the set of smooth points in Z is complex analytic in M. Will it follow that Z is complex analytic?... • 8,323 2 votes 0 answers 45 views ### Semilinear elliptic equations in complex plane Let D denote the closed unit disk centered at the origin in the complex plane. Let F: D \times \mathbb C \to \mathbb C be a smooth function. Is there any theory for well-posedness (in the sense of ... • 3,107 3 votes 3 answers 174 views ### Evaluating the series \sum_{n=0}^{\infty} n! x^n and inverse variable-fractional-derivatives So I was interested in formally assigning values to the completely divergent series G(x) = \sum_{n=0}^{\infty} n!x^n . I guess the question COULD end here if you already have an idea of how to ... 13 votes 3 answers 1k views ### Is anything known about the series \sum_{n=0}^{\infty} x^{\sqrt{n}} ? It's well known that there are a shocking number of identities for the usual Jacobi theta function$$ \theta_3(x) = \sum_{n=-\infty}^{\infty} x^{n^2}. $$So I wanted to turn my attention to slowly ... 1 vote 0 answers 122 views ### What can be said about cluster sets for power series of two variables? I'm still trying to prove the continuity of a function u which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ... 0 votes 1 answer 117 views ### Seeking an integral formulation for an algebraic function While working with a generating function for the Catalan numbers, I came across the integral representation$$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\... 2 votes 0 answers 84 views ### The dual of the Lefschetz operator under a perturbation Let$(X, \omega)$be a compact Kähler, or more generally, Hermitian manifold. Let$L_{\omega} : \Omega^k(X) \to \Omega^{k+2}(X)$denote the Lefschetz operator given by $$L_{\omega}(\alpha) : = \omega \... • 137 4 votes 2 answers 278 views ### Borel summation and the Abel function of e^z-1 This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,... 0 votes 0 answers 74 views ### Construction of an L^p function Is possible to construct a function \chi defined in unit ball of \mathbb{C}^n which is in L^p for p large such that (f\ast \chi)(z)\geq f(z), where f\in L^1 ? 2 votes 1 answer 131 views ### Regularity of boundary of a level set of a C^{1,\alpha} function Let f:\mathbb{R}^2\to\mathbb{R} be a C^{1,\alpha} function. Denote S_C=\{x\in\mathbb{R}^2\mid f(x)=C \} the level set of f with value C. What i want to ask is, if S_C is nonempty for some ... • 337 3 votes 1 answer 278 views ### Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set? Say we have a power series of two variables, with an associated function f defined as$$ \begin{split} f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\ & a_{n,m} \geq 0 \quad \forall n, m \in\... 2 votes 0 answers 122 views ### "Circulant-Vandermonde" matrix: in search of a formula An$n\times n$circulant matrix$\mathbf{X}_nhas the form \begin{align} \mathbf{X}_n= \begin{bmatrix} x_1 & x_2 & \cdots & x_{n-1} & x_n \\ x_2 & x_3 & \cdots & x_n&... -1 votes 1 answer 217 views ### Significance ofN_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$Let$N(T)$be the number of zeros of Riemann zeta function upto height$T$in the critical strip and$N_0(T)$be the number of zeros on the critical line. What will be the significance of proving ... 3 votes 0 answers 56 views ### Boundary behavior of conformal map on domain satisfying an exterior sphere condition I'm in the middle of a project concerning a Bernoulli-type free boundary problem in$\mathbb{R}^2$and, as part of this project, I would like to understand the boundary behavior of conformal maps on ... • 513 3 votes 1 answer 117 views ### On well separated circular regions in the Riemann sphere and complex polynomials It started with a conjecture I had, see A statement on complex polynomials, which was false for$n \geq 3$, as shown by Noam D. Elkies in his answer there. The present post is an attempt to salvage ... • 3,773 0 votes 0 answers 57 views ### When a strictly positive log pluriharmonic function$g$is equal to the norm of holomorphic function? Suppose$V$is a local analytic variety (singular). Suppose$g$a strictly positive log pluriharmonic function on$V$, i.e.$\log g$is pluriharmonic. I wonder when$g=|f|^2$, where$f$is a ... • 603 1 vote 2 answers 104 views ###$\log(t)$term in the small time expansion of$\mathrm{Tr}( A e^{-tB} )$Assume$A$is an operator on a Hilbert space with discrete spectrum. Assume$B$is a positive operator on the same Hilbert space also with a discrete spectrum. Also assume$A$and$B$commute. I'm ... 5 votes 1 answer 771 views ### A statement on complex polynomials I have a feeling the following is true. Assume that there are$n$mutually disjoint closed disks$D_i$in the complex plane and$n$complex polynomials$p_i(z)$of degree$n - 1$, with both types of ... • 3,773 5 votes 0 answers 199 views ### Picard-Lefschetz formula for the quotient of a degenerating family of curves by a cyclic group$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$(This is a slight rephrasing and modification of the original question) Let$D\subset\mathbb{C}$be the complex unit disk. Let$X$be a ... • 9,220 2 votes 1 answer 100 views ### Finding the repelling fixed point of an exponential, knowing only its attracting one This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult ... 5 votes 1 answer 349 views ### Example illustrating necessity of considering birational equivalence and not biholomorphic equivalence in MMP The minimal model program attempts to classify algebraic varieties up to birational equivalence. For compact Riemann surfaces, Riemann's uniformization theorem tells us that the geometry of the curve ... • 255 9 votes 2 answers 1k views ### Zeros of a complex function I wonder whether$\sum_{k=0}^n \exp(r_k z)$has a complex zero for any$n\in \mathbb{Z}_n^*,0=r_0<r_1<r_2<\dotsb<r_n$. I think the answer is affirmative. • 93 1 vote 2 answers 230 views ### Abscissa of convergence for a very specific Dirichlet series / Euler product I am interested in the convergence of the following Euler product: $$\prod_p \frac{1}{1-\chi(p)\cdot p^{-s}}.$$ The product is over all primes (in increasing order), with$\chi(p)=+1$if$p \bmod 4 =...
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Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that $$\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$ for all \$\tau \...