
Complex analysis graph
I would like to graph these. But I am technologically challenged. Would anybody graph these interesting surfaces.
Graph,
$\displaystyle f(x,y) = e^{\frac{x}{x^2+y^2}}\cos \left( \frac{y}{x^2+y^2} \right)$
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 $\displaystyle (0,0)$ the surface will take every single value possible! Try it, draw the circle $\displaystyle x^2+y^2=1$. Inside this circle $\displaystyle f(x,y)$ takes on every single value.
Graph,
$\displaystyle f(x,y) = \cosh^2y  \sin^2x  1$
What is going on? This is an illustration of something called MaximumModulos 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, $\displaystyle (x1)^2/4^2 + (y+1)^2/6^2 = 1$ on this ellipse the maximum value is on the boundary.