The question:

For the mapping f(z) = sinh(z), find and sketch the image of Re(z) = c

My attempt:

Unless I'm mistaken, Re(z) = C is just a line.

I'm having problems with the f(z) = sinh(z) part. I set f(z) = w, and used the definition of sinh(z) as follows:

$\displaystyle w=sinh(z)$

$\displaystyle w = \frac{e^z - e^{-z}}{2}$

$\displaystyle w = \frac{e^{x+yi} - e^{-(x+yi)}}{2}$

$\displaystyle w = e^x(cos(y) + isin(y)) - e^{-x}(cos(y) - isin(y))$

$\displaystyle w = cos(y)(e^x - e^{-x}) + isin(y)(e^x+e^{-x})$

Now I'm stuck. :/ Have I attempted this correctly, and if so, how do I proceed? Any help would be greatly appreciated!