# Complex Analysis: Difference between the image of the Joukowski-Mapping and $\sin z$ or $\cos z$?

i am currently studying complex analysis, conformal mappings in particular and i am trying to get a good grasp about the different mapping properties of elementary functions such as $$e^z, \log z, \sin z, \cos z$$ etc.

I just recently learned about the Joukowski-Mapping which is $$J(z) = \frac{1}{2}\left(z+\frac{1}{z}\right)$$ which looks very similar to $$\cos z$$.

However, when i looked at the images of the mentioned mappings, i noticed basically no difference between the Joukowski-Mapping and the images of $$\sin z$$ or $$\cos z$$. Is there even any difference whatsoever?

How can i distinguish them from another if i am given the picture of each for example? Can anyone helpe me on this?

Thank you very much for any help!

• What lines/regions were you applying the mapping too? For instance, consider why $J$ leaves the lines parallel to the coordinate axes almost undistorted for sufficiently large $x$ or $y$ (think about the derivative). On the other hand, take $\sin(x + i y) = \sin x \cosh y + i \cos x \sinh y$, fix $x$ and start increasing $y$. Consider what the ratio of the imaginary and real parts tends to and therefore approximately what curve this corresponds to. Also, fix a large $y$ and start varying $x$. Approximately what curve does this give? – Maxim Feb 6 at 17:14