I'll try and help you out, I've never really used cylindrical co-ordinates but I do know that they're defined by three properties; the radial distance from the z-axis, the azimuth (angle to) of the point and height z.

Your equation would result in a straight line on the z-axis since the distance from the z-axis isn't mentioned. Are you sure the equation is correct as it clearly doesn't correspond with the diagram.

Try it, let , and on this grapher, the result is just a straight line.

surf_graph_cylin