Based on your observations in Part 2, formulate conjectures about the behavior of the magnitude of a differentiable function near each of the three types of isolated singularities -- removable singularity, pole, and essential singularity.
Define a new function with a removable singularity at z = 0. The function h defined by
h(z) = (1 - cos(z))/z2
will work. Use the series command to demonstrate that it is removable. Check that your conjecture holds.
Define a new function with a pole at z = 0. Use the series command to demonstrate that it is a pole. Check that your conjecture holds.
Define a new function with an essential singularity at z = 0. How do you know the singularity is essential? Check that your conjecture holds.
Modify your conjectures, if necessary, and then check again. Continue until you are sure your conjectures are true and give a good description of the behaviors of the magnitude of a differentiable function near an isolated singularity.
