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Math Help - Trigonomentic mapping question

  1. #1
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    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
    w = \frac{e^z - e^{-z}}{2}
    2w =  e^z - e^{-z}

    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.
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  2. #2
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    Re: Trigonomentic mapping question

    \displaystyle \sinh{(z)} = \sinh{(x + iy)} = \sinh{(x)}\cos{(y)} + i\cosh{(x)}\sin{(y)}.

    You need to plot the imaginary part of this equal to \displaystyle d, 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*}

    Now you need to plot this.
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