Id' appreciate help understanding why the integral
$ \int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx $
is convergent provided $\lambda > -2$, where $\phi \in \mathcal{D}(\mathbb{R})$.
To provide some context: this integral arises in the regularization of the (divergent) integral of $x^{\lambda}_+$ ; i.e.
$ \langle x^{\lambda}_+ , \phi \rangle = \int_0^\infty x^{\lambda} \phi \: dx $
By analytic continuation, this integral can be expressed as
$ \int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx + \int_1^\infty x^{\lambda} \: \phi(x) \: dx \: + \: \frac{\phi(0)}{\lambda + 1} $
The following texts all state that the first integral is convergent provided $\lambda > -2$, but its not obvious to me how it does.
- Generalized Functions, Volume 1 by Gelfand and Shilov (1964) -- page 47 & 48
- Theory of Distributions by M. A. Al-Gwaiz (1992), page 64
- Asymptotic approximation of integrals by R. Wong (2001) -- page 258