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6 votes
1 answer
408 views

On an asymptotic integral decay

Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that $$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$ for all $\lambda > 0$. Does it follow that $...
Ali's user avatar
  • 4,135
1 vote
0 answers
196 views

Asymptotic of a functional as $x\rightarrow \infty$

Consider the following functional : $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...
bambi's user avatar
  • 375
0 votes
0 answers
158 views

On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function

Let $$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function. What are the reasonable asymptotic estimates for $I(T)...
user avatar
1 vote
0 answers
108 views

Asymptotics of an integral by two methods

This was asked in MSE, here, but the answer was not satisfactory. I want to compute the asymptotic behavior of the integral $$ f(K,a)=\int_0^1 (1-x)^Ke^{iKa\frac{x}{1-x}}x^2dx$$ when $K$ is large ...
thedude's user avatar
  • 1,549