Consider the transformation w = (1 + i)z + 3 - 4i. Consider the rectangle ABCD with vertices: A: 1 + i; B: -1 + i; C: -1 - i; D: 1 - i. Do the following: (a) Find the image A'B'C'D' of this rectangle under this transformation. Plot both rectangles on the attached graph paper. (c) How do the areas of the two rectangles compare? Describe the transformation geometrically. (e) What are the fixed points of the transformation?
Added by Ronald A.
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The transformation is given by: $$ w = (1 + i)z + 3 - 4i $$ Let's find the image of each vertex: A'(1 + i): $$ w = (1 + i)(1 + i) + 3 - 4i = 1 + 2i - 1 + 3 - 4i = 3 - 2i $$ B'(-1 + i): $$ w = (1 + i)(-1 + i) + 3 - 4i = -1 + 2 + i - i + 3 - 4i = 4 - Show more…
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Consider the transformation w = (1 + i)z + 3 - 4i. Consider the rectangle ABCD with vertices: A: 1 + i; B: -1 + i; C: -1 - i; D: 1 - i. Do the following: (a) Find the image A'B'C'D' of this rectangle under this transformation. (b) Plot both rectangles on the attached graph paper. (c) How do the areas of the two rectangles compare? (d) Describe the transformation geometrically. (e) What are the fixed points of the transformation?
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a) Sketch the region onto which the sector r ≤ 1, 0 ≤ θ≤ π/4 is mapped by the transformation w=z^4 b) Compare this result to the transformation of w=z^2 c) Use your comparison to predict the effect of the transformation w=z^6
From Rogawski Zeta section 16.6, exercise 29. Let D = f(R), which means that D is the image of the region R under the transformation f. If the transformation is f(u,v) = (u^2, u + v) and R is the region [1, 9] x [0, 9], calculate: Note: It is not necessary to describe D.
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