Originally Posted by

**davismj** $\displaystyle f(z) = \frac{z^2}{sin z}$ with $\displaystyle z_0 = n\pi$ for $\displaystyle n = 0, \pm 1, \pm 2, . . .$

My reasoning is that clearly the function is undefined at the above defined singularities. Some of these are bounded, some of them are not.

Look for the spider (Yeah, well he's sideways in the plot below) or one of his close cousins. Here's the Mathematica code to draw the essential singularity of $\displaystyle \sin(1/z)$. Try and understand why this code does what it does. Now substitute your function: no spider or even one of his relatives so no essential singularity.

Code:

spiderSingularity = ContourPlot[
Re[Sin[1/(x + I*y)]] == 5,
{x, -0.1, 0.1}, {y, -0.1, 0.1},
ContourStyle -> Red, PlotPoints -> 50]