= Fourier transform of a function with reciprocal tails
Suppose $w$ is locally integrable and
$$
w(x)=\frac{c_+}{x}+O(x^{-2})\quad(x\to+\infty),
\qquad
w(x)=\frac{c_-}{x}+O(x^{-2})\quad(x\to-\infty).
$$
Then $\widehat w(\lambda)+(c_+-c_-)\log|\lambda|$ has finite limits at zero from both sides. With the convention $\widehat w(\lambda)=\int e^{-i\lambda x}w(x)dx$, the positive-frequency limit minus the negative-frequency limit is
$$
-i\pi(c_++c_-).
$$
Subtracting suitable reflected copies of $x^{-1}\mathbf1_{(1,\infty)}(x)$ leaves an $L^1$ function, whose Fourier transform is continuous; the jump then follows from the <Dirichlet integral>.
Back to article page
