MAT334--2020F > Chapter 1

Power of Complex Numbers with Arguments Hard to Determine Directly

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Yifei Hu:
Question: let w= 2-i , find w^3 + w
Are we suppose to do multiplications directly or are we suppose to use Euler's formula? Since in this case, \theta = arctan(-1/2), we can't directly come out the sin and cos of n \theta.
Are there any other alternative methods to apply to such complex numbers with a general arguments that can take advantage of Euler's formula's easy computations of power? Can we give the answer to this question as a polynomial of e^iarctan(c)?

Victor Ivrii:
In this case usual cube of the sum would be the most efficient solution