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Question: use the taylor series formula for the exponential function to...

Question details
  • Use the Taylor series formula for the exponential function to show that LaTeX: \cos \theta + i \sin\theta = e^{i\theta}
  • Define LaTeX: \bar z =\overline{x+iy}=x-iy and show that the map LaTeX: z\mapsto \bar z is a reflection about the x-axis.
  • Show that the Euclidean length of a vector is given by LaTeX: |z|=\sqrt{z\bar z}
  • Show that the map LaTeX: z\mapsto e^{i\theta} z does not change the length of vectors.
  • Explain why this map is a rotation by angle LaTeX: \theta. Is it clockwise or anticlockwise?
  • Explain why LaTeX: z\mapsto z+ w (where LaTeX: w is a fixed complex number) is a translation.
  • Discuss how to write a glide transformation in the complex plane.
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