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Complex Analysis - Markushevich's book1 Prob 2.23, 2.24: Complex Algebra
Автор: MathWU悟数
Загружено: 2025-12-23
Просмотров: 4
Описание:
Theory of Functions of a Complex Variable (Vol 1) - A.I. Markushevich
I.2 Complex Numbers
5. Complex Algebra
6. Powers and Roots of Complex numbers
Prob 2.23: Calculate the following quantities:
a) (1 + i)^25;
b) ((1 + i sqrt{3})/(1 - i))^30;
c) (1 - (sqrt{3} - i)/2)^24;
d) (-1 + i sqrt{3})^15 /(1 - i)^20 + (-1 - i sqrt{3})^15 /(1 + i)^20.
Prob 2.24: Prove that
a) (1 + i)^n = 2^{n/2} (cos(n pi/4) + i sin(n pi/4));
b) (sqrt{3} - 1)^n = 2^n (cos(n pi/6) - i sin(n pi/6));
c) (1 + cos(a) + i sin(a))^n = 2^n cos^n(a/2) (cos(na/2) + i sin(na/2)), where a is a real number.
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