Homogeneous distribution

In one dimension, the function

${\displaystyle x_{+}^{\alpha }={\begin{cases}x^{\alpha }&{\text{if }}x>0\\0&{\text{otherwise}}\end{cases}}}$

is locally integrable on R \ {0}, and thus defines a distribution. The distribution is homogeneous of degree α. Similarly ${\displaystyle x_{-}^{\alpha }=(-x)_{+}^{\alpha }}$ and ${\displaystyle |x|^{\alpha }=x_{+}^{\alpha }+x_{-}^{\alpha }}$ are homogeneous distributions of degree α.

However, each of these distributions is only locally integrable on all of R provided Re(α) > −1. But although the function ${\displaystyle x_{+}^{\alpha }}$ naively defined by the above formula fails to be locally integrable for Re α ≤ −1, the mapping

${\displaystyle \alpha \mapsto x_{+}^{\alpha }}$

is a holomorphic function from the right half-plane to the topological vector space of tempered distributions. It admits a unique meromorphic extension with simple poles at each negative integer α = −1, −2, .... The resulting extension is homogeneous of degree α, provided α is not a negative integer, since on the one hand the relation

${\displaystyle x_{+}^{\alpha }[\varphi \circ \mu _{t}]=t^{\alpha +1}x_{+}^{\alpha }[\varphi ]}$

holds and is holomorphic in α > 0. On the other hand, both sides extend meromorphically in α, and so remain equal throughout the domain of definition.

Throughout the domain of definition, x^α
₊ also satisfies the following properties:

• ${\displaystyle {\frac {d}{dx}}x_{+}^{\alpha }=\alpha x_{+}^{\alpha -1}}$
• ${\displaystyle xx_{+}^{\alpha }=x_{+}^{\alpha +1}}$

There are several distinct ways to extend the definition of power functions to homogeneous distributions on R at the negative integers.

χ^α
₊

The poles in x^α
₊ at the negative integers can be removed by renormalizing. Put

${\displaystyle \chi _{+}^{\alpha }={\frac {x_{+}^{\alpha }}{\Gamma (1+\alpha )}}.}$

This is an entire function of α. At the negative integers,

${\displaystyle \chi _{+}^{-k}=\delta ^{(k-1)}.}$

The distributions ${\displaystyle \chi _{+}^{a}}$ have the properties

• ${\displaystyle {\frac {d}{dx}}\chi _{+}^{\alpha }=\chi _{+}^{\alpha -1}}$
• ${\displaystyle x\chi _{+}^{\alpha }=\alpha \chi _{+}^{\alpha +1}.}$

${\displaystyle {\underline {x}}^{k}}$

A second approach is to define the distribution ${\displaystyle {\underline {x}}^{-k}}$, for k = 1, 2, ...,

${\displaystyle {\underline {x}}^{-k}={\frac {(-1)^{k-1}}{(k-1)!}}{\frac {d^{k}}{dx^{k}}}\log |x|.}$

These clearly retain the original properties of power functions:

• ${\displaystyle {\frac {d}{dx}}{\underline {x}}^{-k}=-k{\underline {x}}^{-k-1}}$
• ${\displaystyle x{\underline {x}}^{-k}={\underline {x}}^{-k+1},\quad {\text{if }}k>1.}$

These distributions are also characterized by their action on test functions

${\displaystyle {\underline {x}}^{-k}=\int _{-\infty }^{\infty }{\frac {\phi (x)-\sum _{j=0}^{k-1}x^{j}\phi ^{(j)}(0)/j!}{x^{k}}}\,dx,}$

and so generalize the Cauchy principal value distribution of 1/x that arises in the Hilbert transform.

(x ± i0)^α

Another homogeneous distribution is given by the distributional limit

${\displaystyle (x+i0)^{\alpha }=\lim _{\epsilon \downarrow 0}(x+i\epsilon )^{\alpha }.}$

That is, acting on test functions

${\displaystyle (x+i0)^{\alpha }[\varphi ]=\lim _{\epsilon \downarrow 0}\int _{\mathbb {R} }(x+i\epsilon )^{\alpha }\varphi (x)\,dx.}$

The branch of the logarithm is chosen to be single-valued in the upper half-plane and to agree with the natural log along the positive real axis. As the limit of entire functions, (x + i0)^α[φ] is an entire function of α. Similarly,

${\displaystyle (x-i0)^{\alpha }=\lim _{\epsilon \downarrow 0}(x-i\epsilon )^{\alpha }}$

is also a well-defined distribution for all α

When Re α > 0,

${\displaystyle (x\pm i0)^{\alpha }=x_{+}^{\alpha }+e^{\pm i\pi \alpha }x_{-}^{\alpha },}$

which then holds by analytic continuation whenever α is not a negative integer. By the permanence of functional relations,

${\displaystyle {\frac {d}{dx}}(x\pm i0)^{\alpha }=\alpha (x\pm i0)^{\alpha -1}.}$

At the negative integers, the identity holds (at the level of distributions on R \ {0})

${\displaystyle (x\pm i0)^{-k}=x_{+}^{-k}+(-1)^{k}x_{-}^{-k}\pm \pi i(-1)^{k}{\frac {\delta ^{(k-1)}}{(k-1)!}},}$

and the singularities cancel to give a well-defined distribution on R. The average of the two distributions agrees with ${\displaystyle {\underline {x}}^{-k}}$:

${\displaystyle {\frac {(x+i0)^{-k}+(x-i0)^{-k}}{2}}={\underline {x}}^{-k}.}$

The difference of the two distributions is a multiple of the delta function:

${\displaystyle (x+i0)^{-k}-(x-i0)^{-k}=2\pi i(-1)^{k}{\frac {\delta ^{(k-1)}}{(k-1)!}},}$

which is known as the Plemelj jump relation.

The following classification theorem holds (Gel'fand & Shilov 1966, §3.11). Let S be a distribution homogeneous of degree α on R \ {0}. Then ${\displaystyle S=ax_{+}^{\alpha }+bx_{-}^{\alpha }}$ for some constants a, b. Any distribution S on R homogeneous of degree α ≠ −1, −2, ... is of this form as well. As a result, every homogeneous distribution of degree α ≠ −1, −2, ... on R \ {0} extends to R.

Finally, homogeneous distributions of degree −k, a negative integer, on R are all of the form:

${\displaystyle a{\underline {x}}^{-k}+b\delta ^{(k-1)}.}$