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Math Help - Mapping - Hyperbolic Functions

  1. #1
    xaz
    xaz is offline
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    Mapping - Hyperbolic Functions

    For the mapping f(z) = sinh(z), find and sketch the image of im(z) = d

    z = x + iy
    w = u + iv


    This is the solution given:
    \left(\frac{v}{sin d}\right)^2 - \left(\frac{u}{cos d}\right)^2 = 1 << got this one
    \frac{v}{sin(d)}\geq1\  if\  d\neq k\pi/2 \ for\ k\in\mathbb{Z}
    v=0\ if\ d=n\pi\ for\ n\in\mathbb{Z}
    \{iv:v\geq1\}\ for\ d=(2n+1)\pi/2\ and\ n\in\mathbb{Z}\ even
    \{iv:v\leq-1\}\ for\ d=(2n+1)\pi/2\ and\ n\in\mathbb{Z}\ odd

    But confused about the rest..Please help. Thanks!
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  2. #2
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    Re: Mapping - Hyperbolic Functions

    Quote Originally Posted by xaz View Post
    For the mapping f(z) = sinh(z), find and sketch the image of im(z) = d

    z = x + iy
    w = u + iv


    This is the solution given:
    \left(\frac{v}{sin d}\right)^2 - \left(\frac{u}{cos d}\right)^2 = 1 << got this one
    \frac{v}{sin(d)}\geq1\  if\  d\neq k\pi/2 \ for\ k\in\mathbb{Z}
    v=0\ if\ d=n\pi\ for\ n\in\mathbb{Z}
    \{iv:v\geq1\}\ for\ d=(2n+1)\pi/2\ and\ n\in\mathbb{Z}\ even
    \{iv:v\leq-1\}\ for\ d=(2n+1)\pi/2\ and\ n\in\mathbb{Z}\ odd

    But confused about the rest..Please help. Thanks!
    $w=\sinh(z)=\sinh(x+\imath y)=\sinh(x)\cos(y) + \imath \cosh(x)\sin(y)$

    $u=\sinh(x)\cos(y)$

    $v=\cosh(x)\sin(y)$

    The image of $\Im(z)=d$ is just that of $y=d$ or

    $u=\sinh(x)\cos(d)$

    $v=\cosh(x)\sin(d)$

    you can play with sketching the image of the line $y=d$
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