In Kerson Huang's Quantum Field Theory From Operators to Path Integrals (Amazon link), pages 28 and 29, he calculates the propagator in the following case: time-like, space-like and light-like. First he integrates the time-component of $k$, and arrive this expression: $$ \Delta_F(x)=\frac{i}{4\pi^2}\int_0^\infty dk\,\frac{k^2}{\omega_k}\frac{\sin kr}{kr}e^{i\omega_k|t|} $$
Then he gets the Bessel function in the time-like and the space-like case:
By Lorentz invariance $\Delta_F(x)$ can only depend on $$s\equiv x^2=t^2-\mathbf r^2\tag{2.83}$$ For $s>0$, we can put $\mathbf r=0$ to obtain the representation $$ \Delta_F(x)=\frac{i}{4\pi^2}\int_0^\infty dk\frac{k^2}{\omega_k}e^{i\omega_k\sqrt{s}}=\frac{m}{8\pi\sqrt{s}}H_1^{(1)}(m\sqrt{s})\tag{2.84} $$ For $s<=0$, we put $t=0$ to obtain $$ \Delta_F(x)=\frac{i}{4\pi^2}\int_0^\infty dk\frac{k^2}{\omega_k}\frac{\sin k\sqrt{-s}}{k\sqrt{-s}}=-\frac{im}{4\pi^2\sqrt{s}}K_1(m\sqrt{-s})\tag{2.85} $$
At last, he gets the result in light-like case: a delta function:
where $H_1^{(1)}$ and $K_1$ are Bessel functions. In the time-like gregion $s>0$ the function describes an outgoing wave for large $s$. This corresponds to the $i\eta$ prescription in (2.80). The $-i\eta$ prescription would have yielded an incoming wave. In the space-like region $s<0$ it damps exponentially for large $|s|$. On the light cone $s=0$ there is a delta function singularity not covered by the above formulas: $$\lim_{x^2\to0}\Delta_F(x)=-\frac{1}{4\pi}\delta(x^2)$$
Can anyone explain how he obtained the delta function? I don't understand the limit $s=0$ because the Bessel function is divergent there.