Complex Variable Question?
Hey people got a difficult question here hope someone can help me out!
Find all values of (i-1)^2i and express them in polar form or in the form x + iy with real numbers x and y!
Any help would be appreciated thanks!
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Verified answer
Therefore infinitely many such values.
(-1 + i)^(2i)
= exp [log((-1 + i)^(2i))]
= exp [2i * log(-1 + i)]
= exp [2i * (ln |-1 + i| + i arg(-1 + i))]
= exp [2i (ln(√2) + i (3π/4 + 2πk)] for any integer k
= exp [2i ((1/2) ln 2 + i (3π/4 + 2πk)]
= exp [i ln 2 - (3π/2 + 4πk)]
= exp [-(3π/2 + 4πk)] * exp(i ln 2)
= exp [-(3π/2 + 4πk)] * [cos(ln 2) + i sin(ln 2)] for any integer k.
I hope this helps!
(1-i)^2i=exp(2iln(1-i))
|1-i|=â2
Arg(1-i)=-Ï/4 for the arguments of the complex numbers taken
in ]-Ï,Ï].
(1-i)^2i=exp(2i(lnâ2-iÏ/4))
=exp(2ilnâ2)exp(Ï/2)
(1-i)^2i=exp(iln2)exp(Ï/2)