Laurent series of algebraic function Consider function
$$f(z)=\frac{1}{\sqrt{a^2-z^2}+b\sqrt{c^2-z^2}}.$$
It has pole
$$z = \frac{\sqrt{b^2 c^2 - a^2}}{\sqrt{b^2-1}}$$.
I wanna find Laurent series of $f$ around singular point.
For $b=1$, I put $f$ in partial fraction to proced.
How to find Laurent series  for $b \ne 1$?

Hint:
$$
\frac{1}{\sqrt{a^2-z^2}+b\sqrt{c^2-z^2}}=\frac{\sqrt{a^2-z^2}-b\sqrt{c^2-z^2}}{a^2-z^2-b^2(c^2-z^2)}=\frac{\sqrt{a^2-z_0^2-(z^2-z_0^2)}-b\sqrt{c^2-z_0^2-(z^2-z_0^2)}}{(b^2-1)(z^2-z_0^2)},
$$
where
$$
z_0 = \frac{\sqrt{b^2 c^2 - a^2}}{\sqrt{b^2-1}}.
$$
