Hi Mark
The book is using the following theorem:
If f(x,y) is analytic at (x_0,y_0), i.e if f(x,y) has a Taylor series expansion in powers of (x-x_0) and (y-y_0), valid for |x-x_0|<r,|y-y_0|<r, and if for every (x,y) in this 2rx2r rectangle which has (x_0,y_0) at its center,
|f(x,y|\leq M...