How Does f(z) = z + 1/z Map a Circle to an Ellipse?

[joeblow]

How does the function f(z) = z + 1/z take a circle of radius g.t. 1 to an ellipse? How do I think about it geometrically ? (i.e., how should I be able to look at the complex function and tell straight away)

 

[torquil]

How does the function f(z) = z + 1/z take a circle of radius g.t. 1 to an ellipse? How do I think about it geometrically ? (i.e., how should I be able to look at the complex function and tell straight away)

I can't see it by just looking at it, but I did manage to prove it by inserting $$z=re^{i\phi}$$, then finding x and y-components of the expression f(z) in terms of $$\cos(\phi)$$ and $$\sin(\phi)$$. Then I used the relationship $$cos^2 + sin^2 = 1$$ to find an expression among x and y, which turned out to be the equation for an ellipse, as long as r>=1.