Question

Find the bilinear transformation which maps the points = −1, 0, 1 into
the points = 0,i,3i respectively.​

Answer 1

The bilinear transformation is defined as: z = (az' + b)/(cz' + d)

Where a, b, c, and d are complex numbers. To find the specific values for these numbers, we can use the given information about the points being mapped.

We know that z' = -1 maps to z = 0, so we can set up the following equation:

$0 = (a*-1 + b)/(-1*c + d)$

This can be simplified to:

$0 = a/d - b/d$

So we know that a/d = b/d. We can then let a/d = k.

Next, we know that z' = 0 maps to z = i, so we can set up the following equation:

i = (a0 + b)/(0c + d)

This can be simplified to:

i = b/d

So, we know that b/d = i.

Finally, we know that z' = 1 maps to z = 3i, so we can set up the following equation:

$3i = (a1 + b)/(1c + d)$

This can be simplified to:

$3i = (a + b)/(c + d)$

So we know that (a + b)/(c + d) = 3i.

Putting all of this information together, we can find the values of a, b, c, and d.

Since a/d = k and b/d = i, we know that a = ki and b = i^2. From the last equation,

we know that

(a + b)/(c + d)

= 3i, so c + d

= $(a + b)/3i$

=$(ki + i^2)/3i$

= $i/3.$

So the bilinear transformation that maps the points -1, 0, 1 to 0, i, 3i respectively is:

z = $(kiz' + i^2)/(i/3z' + 1)$

Know more from the following links.

brainly.in/question/16766451

brainly.in/question/4211888

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