Using the Fourier transform we can easily define the fractional Laplacian by $$(-\Delta)^{s/2}f(x)=(|\xi|^s\hat f(\xi))^\vee(x), \ \ f\in C_0^\infty. $$
However, I learned that there is another definition using the principal value of singular integral $$(-\Delta)^{s/2}f(x)=C_{n,s}P.V.\int_{\mathbb{R}^n}{\frac{f(x)-f(y)}{|x-y|^{n+s}}dy}, \ \ 0<s<2, $$ where $C_{n,s}$ is some constant. I am curious about the proof that these two definitions are equivalent, but I can not find the proof on the internet or textbooks. By the way, I found that one may symmetrize the integral above to regularize the integral $$ P.V.\int_{\mathbb{R}^n}{\frac{f(x)-f(y)}{|x-y|^{n+s}}dy}= P.V.\int_{\mathbb{R}^n}{\frac{f(x)-f(x-y)}{|y|^{n+s}}dy}\\=\frac12\int_{|y|\le h}{\frac{2f(x)-f(x+y)-f(x-y)}{|y|^{n+s}}dy}+\int_{|y|\ge h}{\frac{f(x)-f(x-y)}{|y|^{n+s}}dy}.$$