TITLE: Approximation of $\sqrt{ x + y } - \sqrt{ x - y }$
QUESTION [5 upvotes]: I've been struggling to try and find a way to approximate the function:
$\sqrt{ x + y } - \sqrt{ x - y }$
I should mention that $y$ is positive and a small number, so that $0<y<<1$.
What I'm hoping for is to approximate this in such a way that, we have roughly:
$\sqrt{ x + y } - \sqrt{ x - y } \approx (1-y)\sqrt{ x + y }$
There may be some numerical factor in front of this. This could very well be absurd, I'm wondering if this can be done at all. It is crucial that I have this factor $\sqrt{ x + y }$ in the approximation.
I've thought about defining a function $\ f(r)=\sqrt{r}$. Then I could write:
$f(x+y)-f(x-y)$ = $f(x+y)-f(x+y-2y)$
i've tried taking a Taylor expansion but my result isn't working out. Does anyone have some advice?

REPLY [2 votes]: Try this one for size:
$$ f(y) \approx \sqrt{x} \left( \arcsin\left(\frac{y}{x}\right) \right) $$
or 
$$ f(y) \approx \sqrt{x} \left( \arcsin\left(\frac{y}{x}\right)-\frac{1}{24}\left(\arcsin\left(\frac{y}{x}\right)\right)^3 \right) $$
The last one is almost an exact fit throughtout the range of $y$. I got it by setting $y=x \sin(r)$ and perfoming a taylor expansion on $$f(r)=\sqrt{x} \left( \sqrt{1+\sin(r)}-\sqrt{1-\sin(r)} \right) \\ = \sqrt{x} \left(r-\frac{r^3}{24}\right)$$
See the comparison for yourself at Wolfram Alpha
Alternative link to Wolfram, http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427ef6ds3l1c74