Trigonomentic mapping question
The question
For the mapping f(z) = sinh(z), find and sketch the image of Im(z) = d.
If I'm not mistaken, this is just a horizontal line in the z-plane through some constant d. With mapping questions involving Z, I usually try and write f(z) in terms of z, then substitute it into the equation.
However this one has me stumped. I tried:
Let f(z) = w
Now I'm unsure of how to write this in terms of z. Is this the correct approach? If so, how do I progress?
Thank you.
Re: Trigonomentic mapping question
} = \sinh{(x + iy)} = \sinh{(x)}\cos{(y)} + i\cosh{(x)}\sin{(y)})
.
You need to plot the imaginary part of this equal to 
, so
![\displaystyle \begin{align*} \cosh{(x)}\sin{(y)} &= d \\ \sin{(y)} &= \frac{d}{\cosh{(x)}} \\ y &= \arcsin{\left[\frac{d}{\cosh{(x)}}\right]} \end{align*}](http://latex.codecogs.com/png.latex?\displaystyle \begin{align*} \cosh{(x)}\sin{(y)} &= d \\ \sin{(y)} &= \frac{d}{\cosh{(x)}} \\ y &= \arcsin{\left[\frac{d}{\cosh{(x)}}\right]} \end{align*})
Now you need to plot this.
