Cauchy principal value

Distribution theory

Let ${\displaystyle C_c^{\mathrm{\infty }}(\mathbb{R})}$ be the set of test functions, i.e., the space of smooth functions with compact support on the real line ${\displaystyle \mathbb{R}}$. Then the map $${\displaystyle \mathrm{p}.\mathrm{v}.\left(\frac{1}{x}\right):C_c^{\mathrm{\infty }}(\mathbb{R})\to \mathbb{C}}$$ defined via the Cauchy principal value as $${\displaystyle \left[\mathrm{p}.\mathrm{v}.\left(\frac{1}{x}\right)\right](u)=\underset{\epsilon \to 0^{+}}{lim}{\int }_{\mathbb{R}\setminus [-\epsilon ,\epsilon ]}\frac{u(x)}{x}\mathrm{d}x=\underset{\epsilon \to 0^{+}}{lim}{\int }_{\epsilon }^{+\mathrm{\infty }}\frac{u(x)-u(-x)}{x}\mathrm{d}x\text{for }u\in C_c^{\mathrm{\infty }}(\mathbb{R})}$$ is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears, for example, in the Fourier transform of the sign function and the Heaviside step function.

Well-definedness as a distribution

To prove the existence of the limit $${\displaystyle \underset{\epsilon \to 0^{+}}{lim}{\int }_{\epsilon }^{+\mathrm{\infty }}\frac{u(x)-u(-x)}{x}\mathrm{d}x}$$ for a Schwartz function ${\displaystyle u(x)}$, first observe that ${\displaystyle \frac{u(x)-u(-x)}{x}}$ is continuous on ${\displaystyle [0,\mathrm{\infty }),}$ as $${\displaystyle \underset{x\searrow 0}{lim}[u(x)-u(-x)]=0}$$ and hence $${\displaystyle \underset{x\searrow 0}{lim}\frac{u(x)-u(-x)}{x}=\underset{x\searrow 0}{lim}\frac{u^{\prime }(x)+u^{\prime }(-x)}{1}=2u^{\prime }(0),}$$ since ${\displaystyle u^{\prime }(x)}$ is continuous and L'Hopital's rule applies.

Therefore, ${\displaystyle {\int }_0^1\frac{u(x)-u(-x)}{x}\mathrm{d}x}$ exists and by applying the mean value theorem to ${\displaystyle u(x)-u(-x),}$ we get:

${\displaystyle \left|{\int }_0^1\frac{u(x)-u(-x)}{x}\mathrm{d}x\right|\le {\int }_0^1\frac{|u(x)-u(-x)|}{x}\mathrm{d}x\le {\int }_0^1\frac{2x}{x}\underset{x\in [-1,1]}{sup}|u^{\prime }(x)|\mathrm{d}x\le 2\underset{x\in [-1,1]}{sup}|u^{\prime }(x)|.}$

And furthermore:

${\displaystyle \left|{\int }_1^{\mathrm{\infty }}\frac{u(x)-u(-x)}{x}\mathrm{d}x\right|\le 2\underset{x\in \mathbb{R}}{sup}|x\cdot u(x)|\cdot {\int }_1^{\mathrm{\infty }}\frac{\mathrm{d}x}{x^2}=2\underset{x\in \mathbb{R}}{sup}|x\cdot u(x)|,}$

we note that the map $${\displaystyle \mathrm{p}.\mathrm{v}.\left(\frac{1}{x}\right):C_c^{\mathrm{\infty }}(\mathbb{R})\to \mathbb{C}}$$ is bounded by the usual seminorms for Schwartz functions ${\displaystyle u}$. Therefore, this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution.

Note that the proof needs ${\displaystyle u}$ merely to be continuously differentiable in a neighbourhood of 0 and ${\displaystyle xu}$ to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as ${\displaystyle u}$ integrable with compact support and differentiable at 0.

More general definitions

The principal value is the inverse distribution of the function ${\displaystyle x}$ and is almost the only distribution with this property: $${\displaystyle xf=1\iff \mathrm{\exists }K:f=\mathrm{p}.\mathrm{v}.\left(\frac{1}{x}\right)+K\delta ,}$$ where ${\displaystyle K}$ is a constant and ${\displaystyle \delta }$ the Dirac distribution.

In a broader sense, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space ${\displaystyle {\mathbb{R}}^n}$. If ${\displaystyle K}$ has an isolated singularity at the origin, but is an otherwise "nice" function, then the principal-value distribution is defined on compactly supported smooth functions by $${\displaystyle [\mathrm{p}.\mathrm{v}.(K)](f)=\underset{\epsilon \to 0}{lim}{\int }_{{\mathbb{R}}^n\setminus B_{\epsilon }(0)}f(x)K(x)\mathrm{d}x.}$$ Such a limit may not be well defined, or, being well-defined, it may not necessarily define a distribution. It is, however, well-defined if ${\displaystyle K}$ is a continuous homogeneous function of degree ${\displaystyle -n}$ whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.

Examples

Consider the values of two limits: $${\displaystyle \underset{a\to 0+}{lim}\left({\int }_{-1}^{-a}\frac{\mathrm{d}x}{x}+{\int }_a^1\frac{\mathrm{d}x}{x}\right)=0,}$$

This is the Cauchy principal value of the otherwise ill-defined expression $${\displaystyle {\int }_{-1}^1\frac{\mathrm{d}x}{x},\text{ (which gives }-\mathrm{\infty }+\mathrm{\infty }\text{)}.}$$

Similarly, we have $${\displaystyle \underset{a\to \mathrm{\infty }}{lim}{\int }_{-a}^a\frac{2x\mathrm{d}x}{x^2+1}=0,}$$

This is the principal value of the otherwise ill-defined expression $${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{2x\mathrm{d}x}{x^2+1}\text{ (which gives }-\mathrm{\infty }+\mathrm{\infty }\text{)}.}$$

Notation

Different authors use different notations for the Cauchy principal value of a function ${\displaystyle f}$, among others: $${\displaystyle PV\int f(x)\mathrm{d}x,}$$ $${\displaystyle \mathrm{p}.\mathrm{v}.\int f(x)\mathrm{d}x,}$$ $${\displaystyle {\int }_L^{\ast }f(z)\mathrm{d}z,}$$ $${\displaystyle -\int f(x)\mathrm{d}x,}$$ as well as ${\displaystyle P,}$ P.V., ${\displaystyle \mathcal{P},}$ ${\displaystyle P_v,}$ ${\displaystyle (CPV),}$ ${\displaystyle \mathcal{C},}$ and V.P.