I would like to graph these. But I am technologically challenged. Would anybody graph these interesting surfaces.

Graph,
f(x,y) = e^{\frac{x}{x^2+y^2}}\cos \left( \frac{y}{x^2+y^2} \right)

What is going on? This is an illustration of something called Picard's theorem. Which would say that no matter how small you draw a circle around (0,0) the surface will take every single value possible! Try it, draw the circle x^2+y^2=1. Inside this circle f(x,y) takes on every single value.

Graph,
f(x,y) = \cosh^2y - \sin^2x  - 1

What is going on? This is an illustration of something called Maximum-Modulos theorem. Which would say that no matter where you draw a circle the maximum value on that region is attained on the boundary, never inside! Try it. And in fact for any irregular shaped closed curve. Take for example, (x-1)^2/4^2 + (y+1)^2/6^2 = 1 on this ellipse the maximum value is on the boundary.