Answer to Question #173747 in Complex Analysis for Rayhan

Question #173747

Let α be a complex number that satisfies both the equations from the previous question. Which of the following are possible arguments of the cube roots of α?

a) π/6

b) 5π/6

c) π/12

d) π/2

e) 5π/12

f) 0

g) π

h) -π/2










1
Expert's answer
2021-03-23T05:31:32-0400

"\\alpha=r(cos\\varphi+isin\\varphi)"


"\\sqrt[3]{\\alpha}=\\sqrt[3]{r}(cos(\\frac{\\varphi+2k\\pi}{3})+isin(\\frac{\\varphi+2k\\pi}{3}))" , "0\\leq k\\leq 2"

For example:

if "\\varphi=\\pi\/2" then arguments "\\theta" of the cube roots of "\\alpha" :

"k=0" : "\\theta=\\pi\/6"

"k=1" : "\\theta=5\\pi\/6"

"k=2" : "\\theta=3\\pi\/2"

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