I have been reading this text from Michelle Verne on Equivariant Cohomology and stumbled upon this specific paragraph which I can't seem to make sense of:
Let us point out some examples of non-compact manifolds $N$ where the equivariant symplectic volume exists in the sense of generalized functions. We will use the following generalized functions: $$Y^+ (\phi) := \int_0^\infty e^{i\phi t} dt, \qquad Y^- (\phi) := \int_{-\infty}^0 e^{i\phi t} dt, \qquad \delta_0(\phi) := \int_{-\infty}^\infty e^{i\phi t} dt,$$ Note that the generalized function $Y^+$ is the boundary value of the holomorphic function $\frac{1}{-i\phi}$ defined on the upper-half plane, so that it satisfies the relation $(-i\phi)Y^+(\phi) = 1$.
I couldn't understand this last statement for the first generalized function since is not defined. She uses a definition of tempered distribution to make sense of it, though I didn't catch it either.
Has anyone seen this? Any help is appreciated.