Statement about entire functions

Question

In Stein and Shakarchi's Complex Analysis, there is the following statement: if two entire functions, say $f_1 $,$f_2$, vanish at all $z=a_n $ and nowhere else, then $\frac {f_1}{f_2} $ has removable singularities at all the points $a_n $. Suppose $f_1$ has a simple zero in $a_1 $ and $f_2$ has a zero of order $2$ in $a_1$; then $a_1$ would be a pole for $\frac {f_1}{f_2} $. Where is the absurd in this supposition? Thanks for any clarify

Answer 1

The statement in your book is false. Your arguments are correct.

An example: let $f_1(z)= \sin z$ and $f_2(z)= \sin^2 z.$ Then $\frac{f_1}{f_2}(z)= \frac{1}{ \sin z}.$

Each zero of $ \sin$ is a simple pole of $\frac{f_1}{f_2}.$