Examples of poles, removable singularities and essential singularities - verification

Question

Give examples of functions $f_1, f_2, f_3 \in H(D'(a,1))$ which have respectively removable singularity, pole of order 3 and essential singularity.

I though about:

• $f_1 = \frac{z^2 - a^2}{z-a}$
• $f_2 = \frac{1}{(z-a)^3}$
• $f_3 = e^{\frac{1}{z-a}}$

$f_1$ can be removed $ \frac{z^2 - a^2}{z-a} = \frac{(z-a)(z+a)}{z-a} = z+a$, $f_2$ seems like pole of order 3 and using Taylor's series $f_3$ seems to have infinitely many zeros in $z-a$.

Is this correct answer?

Answer 1

• It is correct. Or you can take any constant function, for instance.
• It is correct. It would be simpler to take $\frac1{z-a}$, which has a simple pole at $a$.
• I have no idea about the meaning of “to have infinitely many zeros in $z−a$”. But is is correct, since$$e^{1/(z-a)}=\sum_{n=-\infty}^0\frac{(z-a)^n}{(-n)!}.$$