How to proceed computing $L^1(dA)$ integrals like the following

Question

I am trying to check if the following function is in $L^1(dA)$ where A is the area measure defined on the unit disk

$f(z)=\frac{1}{(1-z)^{3/2}}$

I am trying to check its membership in $L^1(dA)$.

Answer 1

Hint: The only potential problem is near $1.$ You'll be done if you show, say,

$$\int_{D(1,1)}|1-z|^{-3/2}\,dA(z)< \infty.$$

That integral is the same as $\int_{D(0,1)}|w|^{-3/2}\,dA(w).$ Use polar coordinates.