Complex Variables and Partial Differential Equations (3130005)

Question-2b

BE | Semester-3 Summer-2020 | 27-10-2020

Q2) (b)

Determine the Mobius transformation which maps $z = \infty, i, 0$ into $w = 0, i, \infty$ .

Here,

z_{1} = \infty, z_{2} = i, z_{3} = 0; w_{1} = 0, w_{2} = i, w_{3} = \infty .

The cross ratio,

\frac{(w - w_{1}) (w_{2} - w_{3})}{(w - w_{3}) (w_{2} - w_{1})} = \frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})}

\Rightarrow \lim_{w_{3} \to \infty} \frac{(w - 0) (i - w_{3})}{(w - w_{3}) (i - 0)} = \lim_{z_{1} \to \infty} \frac{(z - z_{1}) (i - 0)}{(z - 0) (i - z_{1})}

\Rightarrow \lim_{w_{3} \to \infty} \frac{{ww}_{3} (\frac{i}{w_{3}} - 1)}{{iw}_{3} (\frac{w}{w_{3}} - 1)} = \lim_{z_{1} \to \infty} \frac{{iz}_{1} (\frac{z}{z_{1}} - 1)}{{zz}_{1} (\frac{i}{z_{1}} - 1)}

\Rightarrow \frac{w (0 - 1)}{i (0 - 1)} = \frac{i (0 - 1)}{z (0 - 1)}

\Rightarrow w = \frac{i^{2}}{z}

\Rightarrow w = - \frac{1}{z}

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