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Let g and h be analytic in the open disc [itex]\{z \in \mathbb{C} : |z-a| < r \}[/itex], r>0and let [itex]f(z)=\frac{g(z)}{h(z)}[/itex]

if [itex]g(a) \neq 0, h(a)=0, h'(a) \neq 0[/itex] show that f has a pole at z=a and find the corresponding residue of f at a.

Now I initially though we had to Laurent expand the functions g and h but that got really complicated very quickly and i couldn't find anyway of sorting stuff out.

then i tried taylor expanding them which was nicer but still didn't rearrange well.

so my 2 questions are:

(i) how do i do the above problem, and

(ii) am i correct in saying that you can't taylor expand a complex function you can only laurent expand it and if the negative coefficients turn out to be 0 then it reduces to a taylor expansion or are we allowed to do taylor expansions? if we are allowed to, aren't we ignoring the negative terms - is this allowed?