What is meant by 'stretching the neighbourhood'? In the book I have it is written in an example -

To determine an expansion valid in the boundary layer, we need to stretch the neighbourhood of $x=1$. Thus, we let $\xi=(1-x)/\epsilon^v$.

What is meant by 'stretching the neighbourhood of $x=1$? I somewhat understand the concept of stretching the boundary layer, but what does the quoted part mean mathematically?
The problem is of 'boundary layer problems' of perturbation theory. The boundary layer is at the right end of $[0,1]$.
 A: I like to think of re-scaling in singular perturbation problems as "zooming in".
In this example, you have a small region near $x=1$ where your function changes really quickly. If you zoom in to the region near $x=1$ it will look like it's changing much more slowly. When solving equations, treating both the fast-changing and slow-changing parts together is difficult. But when you have zoomed in you can focus on just the fast-changing part of the solution and can forget about the rest of the solution. This gives you a simpler equation to try and solve. You can then solve for the slowly-changing part of the solution separately, and then finally combine both parts of solution.
Put another way, near $x=1$ your solution changes by $O(1)$ over a region of width $O(\epsilon)$. If you scale your independent variable by $\epsilon^{-1}$ then in the scaled coordinates your solution changes by $O(1)$ over a region of width $O(1)$. You can safely ignore terms that are very small relative to 1 (the $O(\epsilon)$ terms, for example).
