let f(z) be the meromorphic function fiven by function $f(z)=\frac{z}{(1-e^z).\sin z}.$ then choose the True statement
a) For every $ k$ $\in$ Z \ {0} ,$ z= k\pi$ is a simple pole
b) $z= \pi + 2\pi i$ is a pole
My attempts : I think both option a) and b) are True
because for option a) $e^z = 1,$ $ e^z = e^{2n \pi i}$, Now $sink\pi =0$
similarly for option b) $sin( \pi + 2\pi i)= sin\pi = 0$
Is its correct or not ???
$\endgroup$1 Answer
$\begingroup$No, it is not correct. The only zros of the sine function or the integer multiples of $\pi$. In particular, $\sin(\pi+2\pi i)\neq0$.
But, yes, statement a) is true.
$\endgroup$ 2
