Discussion
Which of the following complex functions has a pole at \(z = 0\) ?
Answer: C
The function with a pole at 0 is \(f(z) = \dfrac{1 + z + 2z^{3}}{z^{4} - z^{7}}\).
Factor the denominator: \(z^{4} - z^{7} = z^{4}(1 - z^{3})\). Near 0 the factor \(1 - z^{3}\) is close to 1 and the numerator is close to 1, so
\[ \begin{aligned} f(z) &= \frac{1}{z^{4}} \\ &\quad \cdot \frac{1 + z + 2z^{3}}{1 - z^{3}} \end{aligned} \]
with the second factor analytic and equal to 1 at 0. That is a pole of order 4.
\(e^{1/z^{3}}\) and \(\sin(1/z)\) have Laurent series with infinitely many negative powers of \(z\), so 0 is an essential singularity of each, not a pole. \(z^{3} + 7z + 1\) is a polynomial, analytic everywhere, with no singularity at all.
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