Paper 1, Section I, C

Vectors and Matrices | Part IA, 2018

For z,wCz, w \in \mathbb{C} define the principal value of zwz^{w}. State de Moivre's theorem.

Hence solve the equations (i) z6=3+iz^{6}=\sqrt{3}+i, (ii) z1/6=3+iz^{1 / 6}=\sqrt{3}+i, (iii) iz=3+ii^{z}=\sqrt{3}+i (iv) (e5iπ/2)z=3+i\left(e^{5 i \pi / 2}\right)^{z}=\sqrt{3}+i

[In each expression, the principal value is to be taken.]

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