Math Problem Statement

Find where the M ̈obius transformation f (z) = z + 1 z − i maps the unit circle, the real axis, and the imaginary axis. Also draw graphs of the unit circle and real and imaginary axes in the z-plane and their images in the w-plane. • Note that you only need to calculate the images of 0, −1, 1, −i, i, and ∞ (note that f (∞) = limz→∞ f (z), assuming the limit exists). • Remember that M ̈obius transformations preserve angles.

Solution

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Math Problem Analysis

Mathematical Concepts

Complex Analysis
Möbius Transformation
Conformal Mapping

Formulas

Möbius transformation: f(z) = (z + 1) / (z - i)
Limiting behavior of f(z) as z → ∞

Theorems

Möbius transformations preserve angles
Möbius transformations map circles and lines into circles and lines

Suitable Grade Level

Undergraduate