Note: this post is a follow up to an earlier question.
The (divergent) integral of $x^{\lambda}_+$ can be analytically continued into the region Re $\lambda > -n - 1$, $\lambda \ne -1, -2 , \ldots , -n$ by the expression
$ \begin{align*} \int_0^\infty x^{\lambda} \phi \: dx &= \int_0^1 x^{\lambda} \left[ \phi(x) - \phi(0) - x \phi^\prime(0) - \cdots -\frac{x^{n - 1}}{(n - 1)!}\phi^{(n - 1)}(0) \right] dx \\\ &+ \int_1^\infty x^\lambda \phi(x) dx + \sum_{k = 1}^n \frac{\phi^{(k - 1)}(0)}{(k - 1)! (\lambda + k)}. \end{align*} $
This is achieved by a repeatedly applying the first $n$ terms of the Taylor expansion of $\phi$ at $x = 0$, by a generalization of the method discussed in my earlier question.
According to the texts that I'm studying, in any strip of the form $-n - 1 < \mbox{Re}\lambda < -n$ the above equation can be written in the simple form:
$ \langle x^{\lambda}_+ , \phi \rangle = \int_0^\infty x^{\lambda} \left[ \phi(x) - \phi(0) - x \phi^\prime(0) - \cdots -\frac{x^{n - 1}}{(n - 1)!}\phi^{(n - 1)}(0) \right] dx. $
It seems like a trivial step, as none of the texts that I'm studying bother to explain it; However, I've spent exactly one week trying to figure out this step but I'm nowhere near doing so.
Reference texts:
- Generalized Functions, Volume 1 by Gelfand and Shilov (1964) -- page 47 & 48
- Asymptotic approximation of integrals by R. Wong (2001) -- page 258
- Generalized Functions: theory and technique by Ram P. kanwal (1998) -- page 86