6. find the image of the imaginary axis under the mapping w = z 2

General
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  1. find the image of the imaginary axis under the mapping w = z 2 .
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What is the image of the imaginary axis under the mapping ( w = z^2 )?

Answer: To understand what happens to the imaginary axis under the transformation ( w = z^2 ), let’s break it down step-by-step.

  1. Identify a Point on the Imaginary Axis:

    • A point on the imaginary axis can be represented as ( z = iy ), where ( y ) is a real number and ( i ) is the imaginary unit.

  2. Apply the Mapping ( w = z^2 ):

    • Substitute ( z = iy ) into the mapping:
      $$ w = (iy)^2 $$

  3. Simplify the Expression:

    • Squaring ( iy ) gives:
      $$ w = i^2 \cdot y^2 $$

  4. Calculate ( i^2 ):

    • Recall that ( i^2 = -1 ), so:
      $$ w = -y^2 $$

  5. Interpret the Result:

    • The expression ( w = -y^2 ) is always a real, non-positive number because ( y^2 ) is non-negative for any real ( y ).

Summary: The image of the imaginary axis under the mapping ( w = z^2 ) is the non-positive real axis, or ([- \infty, 0]) on the real number line.

If you have any further questions or need more examples, feel free to ask! @anonymous2