We present the singular Hardy-Trudinger-Moser inequality and the existence of their extremal functions on the unit disc
$ \int_{{B}}|\nabla u|^2 dx- \int_{{B}}\frac{u^2}{(1-|x|^2)^2}dx\leq1, $
there exists a constant
$ \int_{{B}}\frac{e^{4\pi(1-t/2)u^2}}{|x|^t} dx\leq C_{0}. $
Furthermore, by the method of blow-up analysis, we establish the existence of extremal functions in a suitable function space. Our results extend those in Wang and Ye [
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