Sachchidanand Prasad

21-11-2023

Problem: Let the rectangular region $\mathcal{R}$ in the $z$ plane be bounded by $x = 0, y = 0, x = 2, y = 1$. Determine the region $\mathcal{R} ^\prime $ of the $w$ plane into which $\mathcal{R} $ is mapped under the transformations:
  1. $w = z + (1 - 2\iota )$;
  2. $w = \sqrt{2} e^{2\pi \iota / 4}z$;
  3. $\sqrt{2} e^{\pi \iota /4}z + (1 - 2\iota ) $.
Solution: The given rectangular region is
The region R
  • The given transformation is \[ w = z + (1 - 2\iota ) = x + \iota y + (1 - 2\iota ) = (x + 1) + \iota (y-2). \] Therefore,
    • the line $x = 0$ will transform to $u = 0$
    • the line $x = 2$ will transform to $u = 3$
    • the line $y = 0$ will transform to $v = -2$ and
    • the line $y = 1$ will transform to $v = -1$.
    In summary, the given transformation translate each point of the region $\mathcal{R} $ by $1 - 2\iota $ units, and hence the transformed region will be
    R'
  • The given transformation is \[ w = \sqrt{2}e^{\pi \iota /4}z = (x - y) + \iota (x + y) = u + \iota v \] Therefore,
    • the line $x = 0$ will transform to $u = -y$ and $v = y$, that is $u = -v$.
    • The line $x = 2$ will transform to $u = 2 - y$ and $v = 2 + y$, that is $u + v = 4$.
    • The line $y = 0$ will transform to $u = x$ and $v = x$, that is $u = v$.
    • Finally, the line $y = 1$ will transform to $u = x - 1$ and $v = x + 1$, that is, $v - u = 2$.
    In summary, the given transformation rotate each point of the region $\mathcal{R} $ by an angle of $\pi /4$, and stretches the region by $\sqrt{2} $ units. Hence the transformed region will be
    R'
  • The given transformation is \[ w = \sqrt{2}e^{\pi \iota /4}z + (1 - 2\iota ). \] This translates the last region by $1$ unit right and $2$ units down. In particular,
    • the line $x = 0$ will transform $u + v = 1$.
    • The line $x = 2$ will transform to $u + v =3$.
    • The line $y = 0$ will transform to $u - v =3$.
    • Finally, the line $y = 1$ will transform to $u - v =1$.
    Hence the transformed region will be
    R'