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Sektion 3.4

[Matematik/Kalkulus (Lindstroem)/Kapitel 3]

Opgave 1

Find kvadratrødderne til z og skriv dem på polarform og formen a + ib.

a) $$ z = i, \\ r = \sqrt(0^2 + 1^2) = 1, \\ sin \theta = 1 \Rightarrow \theta = \pi/2 \\ z = e^{i\pi/2}, w_0 = e^{i\pi/4} = \sqrt 2/2 + i\sqrt 2/2, \\ w_1 = e^{i5\pi/4} = -\sqrt 2/2 -i\sqrt 2/2 $$

b) $$ z = -i, \\ r = 1, \\ sin \theta = -1 \Rightarrow \theta = 3\pi/2z = e^{i3\pi/2}, \\ w_0 = e^{i3\pi/4} = -\sqrt 2/2 + i\sqrt 2/2, \\ w_1 =e^{i7\pi/4} = \sqrt 2/2 - i\sqrt 2/2 $$

c) $$ z = 2 + 2i\sqrt 3, \\ r = \sqrt(2^2 + (2\sqrt 3)^2) = \sqrt 16 = 4, \\ cos \theta = 1/2 \Rightarrow \theta = \pi/3 \\ z = 4e^{i\pi/3}, w_0 = 2e^{i\pi/6} = \sqrt 3 + i, \\ w_1 = 2e^{i7\pi/6} = -\sqrt 3 - i $$

d) $$z = -1 + i\sqrt 3, \\ r = \sqrt 4 = 2, \\ cos \theta = -1/2 \Rightarrow \theta = 2\pi/3 \\ z = 2e^{i2\pi/3}, \\ w_0 = \sqrt 2e^{i\pi/3} = \sqrt 2/2 + i\sqrt 6/2, \\ w_1 = 2e^{i4\pi/3} = -\sqrt 2/2 - i\sqrt 6/2 $$

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