Discussion
Which of the following complex functions has an infinite number of poles ?
Answer: B
The function with infinitely many poles is \(\sec z\).
\(\sec z = 1/\cos z\) has a simple pole wherever \(\cos z = 0\), that is at \(z = \pi/2 + k\pi\) for every integer \(k\): infinitely many isolated poles.
\(e^{z}\) is entire with no poles. \(\cos(1/z)\) has a single singularity, at 0, and it is essential, not a pole. \(1/(z^{5} + z + 1)\) has poles only at the five roots of the denominator, a finite number.
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