# Tagged Questions

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1answer
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### Questions about expansion of $f(x)=\sum_{i=1}^{\infty} a_i x^i$

In complex field, assume $f(x)=\sum_{i=1}^{\infty} a_i x^i$ where $a_i \in {\bf N}$ or $a_i = 0$, and $f(x)$ converges in an area. Question 1: are there $$f(x)=p(x)+\sum_{i=1}^{\infty}r_i(x),$$ or ...
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### Meromorphic Functions as Distributions

For the function $\frac{1}{x}$ on the real line, one can use a modified principal value integral to consider it as a distribution p.f.$(\frac{1}{x}),$ and one can do a similar construction to make ...
2answers
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### A question from complex analysis

Let $n\geq2$. We assume $0<\alpha_n<\cdots<\alpha_2<\alpha_1<1$ and $0<\beta_n<\cdots<\beta_2<\beta_1<1$ , $\alpha_n=\beta_n$, and there exists $1\leq j_0\leq n$ such ...
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### Polynomial growth of Fourier transforms

I am looking for a theorem that guarantees the polynomial growth of a function $f$ defined by a Fourier integral, that is, when $$f(x)=\int_{-\infty}^{\infty}F(y)e^{ixy}dy.$$ I am only interested in ...
1answer
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### A series question related to solution of Laplace equation

Let $u(x,y)$ be the solution of the Laplace equation $\Delta u=0$ on the unit square $(0,1)\times (0,1)$ with boundary condition: $$u(x,1)=1, u(x,0)=0, u(0,y)=0, u(1,y)=0$$ The series solution is ...
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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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1answer
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### An integral with Gamma functions

I wanted some insights about the integral in equation A.5 (page 19) in this paper, http://arxiv.org/pdf/1301.7182.pdf What is the derivation of this? Is there something more general from where this ...
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### 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: ...
2answers
177 views