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Math Help - Conformal mapping of an infinite strip

  1. #1
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    Conformal mapping of an infinite strip

    Can anyone explain to me why the exponential function maps the strip { z \epsilon \mathbb{C} : - \Pi / 2 < Im z < \Pi / 2 } onto the RHP? Thanks!!
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  2. #2
    Member HappyJoe's Avatar
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    Take a complex number z = x+iy in your infinite strip, so |y| < pi/2. Then you can apply the definition of the complex exponential to get

    exp(z) = exp(x)*(cos(y) + i*sin(y)) [the Euler formula].

    Notice that this complex number has real part exp(x)*cos(y) and imaginary part exp(x)*sin(y). Since we find ourselves in the right halfplane, if the real part is positive, you need only consider the real part. But exp(x) is always positive, and for y in the given range, cos(y) is positive as well, so the real part of exp(z) is positive.
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  3. #3
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    Great, think thats clicked for me! Thankyou!!!!
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