I have a problem with Power-law distribution. In some sources I found that we can define this class of distribution by probability function: $$ p(x) = k \cdot x^{-\alpha}, $$ where $\alpha>1$ and k is constant.
On wikipedia I found another definition: $$ P(X>x) \sim L(x) \cdot x^{-\alpha+1}, $$
where $\alpha >0$ and $L(x)$ is slowly varying function
I would like to show in formal way that these definitions are equivalent.