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### When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...
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For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula: $$f(\zeta) = ... 0answers 1k views ### Meaning of Cauchy integral theorem - the (co)homology viewpoint I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try. In the elementary theory of analytic functions of 1 complex variable, one ... 0answers 674 views ### Analytic continuation of the Dirichlet generating series of the multiplicative partition function Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series: ... 0answers 331 views ### C^\infty function f:{\bf C}\mapsto {\bf C} such that f(z)\in\overline{{\bf Q}(z)} for all z\in {\bf C} Suppose that f:{\bf C}\mapsto {\bf C} is a C^\infty function such that f(z)\in\overline{{\bf Q}(z)} for all z\in {\bf C}, ie f(z) is algebraic over the field {\bf Q}(z) generated by z ... 0answers 64 views ### What does this number tell me about a convex lattice polygon? EDIT: I realized I'd tricked myself by working with a too special case of f, the question is now updated (boundary lattice points replaced vertices). Suppose I have a convex lattice polygon P, ... 0answers 285 views ### Convergence at the radius of convergence Suppose I have (roughly speaking) a multivalued meromorphic function f(z) on all of \mathbb{C} that is single-valued and holomorphic on the open unit disc and has some branch points of finite ... 0answers 581 views ### Hardy spaces: analysis <---> martingales Let H^p be the Hardy space of analytic functions on the open unit disk D: f \in H^p if f is analytic on D and \sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta < \infty. ... 0answers 156 views ### the “three-point” characterization of holomorphy I want to know the source of the following "folkloric" fact about holomorphic functions. It seems well described by the phrase: The three-point characterization of holomorphy. If F is a ... 0answers 135 views ### Perturbations of zero-dimensional algebraic varieties Let P(z,w):\mathbb C^2\to\mathbb C be a certain polynomial, and consider p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C its restriction to the real torus. Assume generic situation, so by ... 0answers 556 views ### some questions about properties of harmonic measure The original post The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle ... 0answers 105 views ### residue and regulator Let C be a curve defined over \mathbb{Q}. The regulator is a map$$ reg: K_2(C)_{\mathbb{Q}} \longrightarrow H^1(C(\mathbb{C}), \mathbb{R}). $$Here K_2(C)_{\mathbb{Q}} is the K-group tensor ... 0answers 270 views ### Etymology of the O-notation for algebras of holomorphic functions The notation O(X) seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in \mathbb{C}^n (or a complex manifold). I would like to know where ... 0answers 289 views ### Laplace Transform: Are there theorems similar to the Bernstein Theorem? Bernstein's Theorem states, that if a function is completely monotonic, then it is the Laplace transform of an L^1-function. (E.g. Widder, "The Laplace Transform", Chapter IV, Theorem 19b) Are ... 0answers 321 views ### Is this an injective function ? Hi all, I got stuck with a problem that pop up in a paper about location of zeros for some analytic functions that I am working on. The problem is the following: Fix two arbitrary positive ... 0answers 803 views ### A “Cauchy integral formula” for the Poisson kernel? The inspiration for this question is in a certain breakdown of the analogy between holomorphic and harmonic functions. First recall the Cauchy integral formula: Let U be an open subset of ... 0answers 326 views ### Adeles of Holomorphic Functions In number theory, an adele is a kind of product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ... 0answers 72 views ### Grunsky-Motzkin-Schoenberg formula I found this formula in Brian McCartin's interesting book "Mysteries of the equilateral triangle" http://www.kettering.edu/news/mysteries-equilateral-triangle and it looks as follows: Suppose that ... 0answers 61 views ### Question concerning Mellin transforms I've recently come across a result I've been trying to generalize. Say that \phi(\sigma \pm iy) < Ce^{\frac{\pi}{2}|y|} in the strip a < \sigma < b then then the following integral is ... 0answers 141 views ### elementary proof for existence of point with minimal period 2 for entire function Fatou proved a very interesting result: for a transcendental entire function f, the second itarate f^{2} has at least has one fixed point. (Using the technique of Picard theorem) This result ... 0answers 109 views ### What is the relationship between complex time singularities and UV fixed points? In this paper it is described how the turbulent kinetic energy spectrum and the flatness (a measure for intermittency) are governed by the position of the (dominant) singularities of the solutions of ... 0answers 246 views ### The functional equation of Hofstadter's Q sequence Hofstadter's Q sequence is defined by Q(1) = Q(2) = 1 and Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2)) for n \geq 3. So far hardly anything on this sequence has been proved -- not even that Q(n) is ... 0answers 135 views ### Bounding an integral transform ouside a circle (or inside a strip) Let g be a symmetric unimodal probability distribution and H be the right half plane. We call$$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$the dispersion function of g. Now, one can ... 0answers 360 views ### Real part of a function on a circle in the complex plane Fix 0 < y <1. Assume that n is a large positive integer. Let H(z):= \sum_{i=1}^n \ln(z^i+i-1). Is it true that \Re(H(e^{\xi+i\theta})) has asymptotically (for large n) a peak at ... 0answers 343 views ### confusion about rational maps and Fatou components Dear fellows, I have come to another conclusion which must be wrong. Let f be a rational map and let U be a connected but not simply connected open subset of the Fatou set such that f(U) is ... 0answers 457 views ### analytic continuation of an integral involving the mittag-leffler function greetings. we have the following integral :$$I(s)=s\int_{0}^{\infty} \frac{dx}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)-\left(E_{s/2}(( x)^{s/2})-1\right)e^{-x}$$where : E_{\alpha}(z) ... 0answers 173 views ### Monotonicity of a certain parametric integral I would like to ask for some help (hints, ideas) in solving the following problem: Given integer n>0 and real \alpha>0,\beta>1 we want to show, that if we define for any ... 0answers 304 views ### Why is Mellin-inverse of Gamma periodic? Specific Case The periodicity is obvious from computation:$$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$However, is there a way to see directly from the integral ... 0answers 408 views ### When checking if a harmonic function is continuous on its boundary, is a dense subset enough? Let U be an open connected subset of \mathbb{C} and let u:U\rightarrow \mathbb{R} be harmonic and bounded on U. Let f:\partial_\infty U \rightarrow \mathbb{R} be a continuous function, ... 0answers 34 views ### Complex function for mapping a circle to a superellipse I was wondering if anyone knows a complex function that would map a circle to a superellipse, or vice versa. Any ideas, comments, or functions are much appreciated! Thanks, Kayvan 0answers 96 views ### Eigenvalue problem I am studying torsional Alfven waves in spicules. In this concern I have encountered the following equation:  \left(1-m^2 e^{-αz}\right)y''(z)+\left(4π i m ... 0answers 118 views ### Questions about transformation or integral transformation I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ... 0answers 242 views ### Separating the eigenvalues of a Hermitian matrix with a special block structure I have a square matrix J \in \mathbb{C}^{2n \times 2n} where, J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix} A \in \mathbb{R}^{n \times n} and is {\bf diagonal}. B \in ... 0answers 217 views ### Convergence of certain L-series Suppose |a_{n}| \leq 1 completely multiplicative function assuming real values. Suppose further that,  L(s)=\sum_{n} \frac{a_{n}}{n^s}  may be continued analytically to the left of s=1 a bit ... 0answers 70 views ### Introducing new poles Suppose we have a contour integral of an entire function along a line in the plane (or perhaps a line segment). I am looking for examples of such integrals that are computed by first altering the ... 0answers 195 views ### An integral with Gamma functions (Part 2) I was wondering if there is a generalization of the integral discussed here to a case like, \int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm ... 0answers 208 views ### Analytical continuation of electrostatic potentials I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the electrostatic ... 0answers 159 views ### Univalent functions with non-negative coefficients Is anything non-trivial known about univalent functions with non-negative coefficients? Let U be the unit disc, and f a univalent (=injective) holomorphic function, f(0)=0, f'(0)>0. ... 0answers 239 views ### How well do continuously differentiable functions behave from R^2 to R^2 ? The behaviour of complex smooth vs 1-dimensional real smooth functions is discussed in a previous question. In "Complex Analysis as Catalyst" by Steven G. Krantz, the Cauchy integral formula is ... 0answers 334 views ### Characterizing essential singularities In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ... 0answers 251 views ### Is the complex harmonic extension of a \mathcal{C^r} map from S^1 to \mathbb{C} is smooth upto the boundary ? Suppose we have a map  h : S^1\to \mathbb{C}  that we know is a \mathcal{C^r}  map ( in the sense of a map between 1-manifold ( or in the sense of a 2\pi periodic map from \mathbb{R}\to ... 0answers 233 views ### Question in complex analysis arising from large N gauge theory This is a question in complex analysis that comes up in the treatment of large N gauge theory with gauge group SU(N) in the 't Hooft limit where N is taken to infinity with \lambda=g^2 N fixed ... 0answers 598 views ### Pole data of meromorphic matrix function Let T(z) be a meromorphic square matrix function, that is - a matrix whose entries are complex meromorphic function of one variable. Recall that such a T is said to have a right pole of order r ... 0answers 222 views ### Finer properties of a sequence of harmonic functions This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer. Background: When ... 0answers 290 views ### What is this effect in Fourier/additive synthesis called? Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was \sum a_{n} sin(2 \pi x * n) and its frequencies were ... 0answers 56 views ### Uniqueness of an embedding theorem for Real differential fields I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let K be a real ... 0answers 41 views ### Determine the position of the contour with the value of corresponding contour integral Let C be the contour of the unit square with lower left corner at origin. We define a function g(z)=\int_{z+C} f(w)dw for a given (not necessarily holomorphic) function ... 0answers 33 views ### singularity of the solution to an integral equation I consider a function x\mapsto f(x) which is the positive solution to the integral equation$$\int_{f(x)}^\infty \frac{du}{u^\gamma-x}=1, \quad x\ge 0, where $\gamma\in (1, 2]$ is some ...
I am trying to compute the asymptotic expansion of the integral $I(t) = \int_{C} e^{\sqrt{1+u}(\frac{1}{t}+\frac{t^2}{\sqrt{u}})}\frac{u^\eta}{\sqrt{1+u}}du$ as $t$ is real and $t\rightarrow +\infty$, ...