exp(1/z)

Here is the picture of exp(1/z). It has an essential singularity at 0. Note the bands of color approaching 0, tangential to the imaginary axis. Recall that very small numbers are colored black and very large numbers colored white. Thus this function is near every complex number in every neighborhood of 0, illustrating our theorem about essential singularities. Actually you can, in this example, verify that w=exp(1/z) is solvable in every neighborhood of 0, so long as w is not 0. This stronger property is also true for essential singularities of any analytic function (Picard's theorem).