- #1

jjhyun90

- 8

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## Homework Statement

A function g is [itex]\alpha[/itex]-regularly varying around zero if for all [itex]\lambda > 0[/itex], [itex]\lim_{x\to 0} \frac{g(\lambda x)}{g(x)}=\lambda^{\alpha}[/itex]

For real s and [itex]\alpha \in (0,1)[/itex], define f:

[itex]f(s)=1-\alpha \int_{0}^{\infty} e^{\alpha t} \frac{\frac{1}{1+s^2}}{e^t(1-\frac{1}{1+s^2})+\frac{1}{1+s^2}} dt = 1 + \alpha \sum_{n=1}^{\infty} \frac{(-s^{-2})^{n}}{n+\alpha}[/itex].

Show f is [itex]2\alpha[/itex]-regularly varying around zero.

## Homework Equations

Note [itex]\frac{1}{1+s^2}[/itex] is a characteristic function of Laplace distribution.

## The Attempt at a Solution

I am not familiar with hypergeometric series. For example, I do not know how to show that the series converges to [itex]-\frac{1}{\alpha}[/itex] when s goes to 0. Are there any properties of hypergeometric series that might be useful for proving this? Attempt to directly compute the limit failed.

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