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if z=i^i^i,where i=(-1)^0.5,then |z| is equal to a)1 b) e^(-pi*/2) c)e^(-pi) d)e^(pi)

if z=i^i^i,where i=(-1)^0.5,then |z| is equal to
a)1
b)e^(-pi*/2)
c)e^(-pi)
d)e^(pi)

Grade:12

1 Answers

Mujtaba Basheer
27 Points
5 years ago
NB: ln i= i*pi/2
z=i^i^i
ln z=( i^i)*ln i
ln z=(i^i)*i*pi/2
ln z= pi/2*i^(i+1)
ln(ln z) = ln(pi/2) + (i+1)*ln i
ln(ln z) = ln(pi/2) + (i+1)*i*pi/2
ln(ln z) = ln(pi/2) - pi/2 + i*pi/2
ln(ln z) = ln(pi/2*e^(-pi/2)*e^(i*pi/2))
ln z= pi/2*e^(-pi/2)*e^(i*pi/2)
ln z = pi/2*e^(-pi/2)*i
ln |z| + i*arg(z) = pi/2*e^(-pi/2)*i
Comparing real and imaginary parts,
ln |z| = 0
|z| = 1

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