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Math Help - Linear mappings of complex function

  1. #1
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    Linear mappings of complex function

    This is from an introductory complex analysis textbook:

    Express the given composition of mappings as a linear mapping f(z)=az+b.

    (A) rotation through \frac{\pi}{4}, magnification by 2, and translation by 1+i
    (B) magnification by 2, translation by \sqrt{2}, and rotation through \frac{\pi}{4}
    (C) translation by \frac{\sqrt{2}}{2}, rotation through \frac{\pi}{4}, then magnification by 2

    The results are all supposed to turn out the same. I get that in (A) the answer is 2e^{i\pi /4}+1+i, but how do you do translations first and then rotation/magnification?
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  2. #2
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    Re: Linear mappings of complex function

    rotation: z = re^iθ → re^i(θ + a) = (e^ia)z , k = (e^ia)
    magnification: z → kz, k real
    translation: z → z + k, k complex

    example: z → (z + k1) → k2(z + k1) → k3[k2(z + k1)]
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