Exercise 20, § 3

Let $n\in \mathbf{N}$. Denote by $V_n$ the open set in $\mathbf{C}$ formed by the complex numbers $\lambda$ such that $\mathscr{R}(\lambda )>-n-1$ and $\lambda \notin \{-1,-2, . . . ,-n\}$.

Let $\lambda \in V_0$. Denote by $x^{\lambda}_+$ the distribution on $\mathbf{R}$ defined by the locally integrable function

$\{0$ if $x\leqslant 0$,

$x^{\lambda}$ if $x >0$.

a) The mapping $\lambda \mapsto x^{\lambda}_+$ of $V_0$ into $\mathscr{D}'(\mathbf{R})$ is analytic. b) Let $n\in \mathbf{N}$. For every $\lambda \in V_n$, the mapping defined on $\mathscr{D}(\mathbf{R})$ by

$\int 1\lambda ('xn-1(n-1))$

$\varphi \mapsto x\varphi (x)-\varphi (0)-x\varphi (0)- \cdots -\varphi$ (0) $dx$

$_0(n-$ 1)!

is a tempered distribution; it does not depend on the integer $n$ such that $\lambda \in V_n$, and coincides with $x^{\lambda}_+$ when $\lambda \in V_0$.

We shall denote by $x^{\lambda}_+$ the distribution resulting in this way for every $\lambda$ belonging to the union of the sets $V_n$ for $n\in \mathbf{N}$.

c) Let $n\in \mathbf{N}$ and let $\lambda \in \mathbf{C}$ such that $-n-1<\mathscr{R}(\lambda )<-n$. For every Schwartz function $\varphi$ on $\mathbf{R}$, one has

$\lambda \int+\infty \lambda ('xn-1(n-1))$

$\langle x_+, \varphi \rangle =x\varphi (x)-\varphi (0)-x\varphi (0)- \cdots -\varphi$ (0) $dx$.

$_0(n-$ 1)!

The mapping from $V_n$ into $\mathscr{D}'(U)$ defined by $\lambda \mapsto x^{\lambda}_+$ is analytic.

d) For every $\lambda$ belonging to the union of the sets $V_m$, one has the formula