2
votes
1answer
314 views

A functional inequality

$g:[0,1]\to[0,1]$ continuously differentiable and increasing such that for all integers $t>0$ and for all $r\in(0,1)$, $g(r^{t+1})>g(r)\cdot g(r^t)$. Does this imply that for all ...
1
vote
0answers
225 views

complex contour integral calculation after Möbius transformation

Good day to everyone. In my scientific research I've got stuck with a contour integration problem. I would like to evaluate the following integral: $$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...
2
votes
1answer
165 views

Limit involving regularlized gamma function and its inverse

Let $$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$ where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function ...
3
votes
3answers
135 views

Multivariate functions whose value is independent of the order of the arguments

Let $r_1, r_2, \ldots, r_k$ be positive integers with or without repetition such that $1\le r_i \le n$ for $i = 1, 2, \ldots, k$. Let $f$ be a continuous multivariate function with the property that ...
2
votes
2answers
291 views

Elliptic function with constant real part on the unit square diagonals?

Consider the following even meromorphic doubly periodic function with poles at the gaussian integer lattice. $H(z) = \prod_{n \in \mathbb{Z}} {1 \over{ 1 - {1 ...
0
votes
3answers
706 views

How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?

$a \in \mathbb{R}$ $f:\mathbb{R} \rightarrow \mathbb{R}$ $g:\mathbb{R} \rightarrow \mathbb{R}$ For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below? $f(x+a)=f(x)+a\times ...
0
votes
0answers
249 views

High dimensional beta integral (question following the previous post)

Hello, This post is a question following the previous post. In one dimensional case, we have $$ \int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} ...
1
vote
2answers
431 views

High dimensional beta integral (a typo in Stein's book “singular integrals”)

Hello, When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake: $$ \int_{R^n} |x-y|^{-n+\alpha} ...