Q1: Let $$f(z) = \sum_{k=-\infty}^\infty a_k z^k$$ be a Laurent series where $a_k = 0$ for all $k<-3$ and $a_{-3} = 1$. Which statement is correct?

A1: $f$ has a pole at $-3$.

A2: $f$ has a pole of order $0$ at $-3$.

A3: $f$ has a pole at $0$ of order $-3$.

A4: $f$ has a pole of order $3$ at $0$.

A5: $f$ has a pole of order $3$ at $1$.

Q2: Let $f(z) = z^2 + \frac{1}{z^3}$. Does $f$ has pole at $0$?

A1: Yes, it’s of order $1$.

A2: Yes, it’s of order $2$.

A3: Yes, it’s of order $3$.

A4: No!

Q3: Which is not a type of an isolated singularity?

A1: removable singularity

A2: linear singularity

A3: essential singularity

A4: pole

Q4: Does $\cos(z^{-1})$ have an isolated singularity at $0$?

A1: Yes, but it’s removable.

A2: Yes, it’s a pole.

A3: Yes, it’s an essential singularity.

A4: No!