Can anyone explain to me why the exponential function maps the strip { : - Im } onto the RHP? Thanks!!

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- Apr 18th 2011, 09:24 AMleshieldsConformal mapping of an infinite strip
Can anyone explain to me why the exponential function maps the strip { : - Im } onto the RHP? Thanks!!

- Apr 18th 2011, 10:29 AMHappyJoe
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. - Apr 19th 2011, 12:36 AMleshields
Great, think thats clicked for me! Thankyou!!!!