While trying assignment problems of complex analysis I am unable to solve this particular question and so I am posting it here.
Let $f$ be a holomorphic function on $0<|z|<\epsilon$ , $\epsilon >0$ given by convergent Laurent series $\sum_{n=-\infty}^{\infty} a_{n} z^{n}$ . Given also that $\lim_{z\to0} |f(z)|= \infty$ .
Then which one is true.
$a_{-1} \neq 0 $ and $a_{-n} =0$ for all $n\ge 2$.
$a_{-N} \neq 0 $ for some $N>1$ and $a_{-n}=0$ for all $n >N$ .
$a_{-n} =0$ for all $n>0$.
$a_{-n} \neq 0$ for all $n>0$.
Although I have studied Ch- Laurent series from text book Complex variables with applications by Ponnusamy and Silvermanbut I am completely clueless on how this particular problem can be approched.
Can anyone please shed some light.