1.If z is a unimodular complex number, prove that
arg(z2 + conjugate of z) = 1/2 arg(z)
2.If \left|z \right| = 1 and z is a non real then prove that z can be expressed in the form of c + \iota /c - \iota where c is real .Also find c in terms of arg z.
3.For c ≥ 1, find all complex numbers z satisfying the equation z + c\left|z+1 \right|+\iota = 0.
4. Prove that \left|z/\left|z \right| -1 \right| \leq \left|argz \right|
11Ans 4)
Put\ Z = r \left(\cos \theta + i\ \sin \theta \right) \\ Thus\ \left|Z \right| = r \\\\ Thus\ \frac{Z}{\left|Z \right|} = \cos \theta + i\ \sin \theta \\\ Hence\ \frac{Z}{\left|Z \right|}\ is\ unimodular
\left|\frac{Z}{|Z|}-\ 1 \right| = | \cos \theta + i\ \sin \theta -1| = 2 |\sin \theta /2| \leq 2 (\theta /2) = \theta = arg\ Z
Thus, Proved
3411) is valid under the assumption that -\frac{\pi}{2} \le \arg z \le \frac{\pi}{2}
Then S = z^2 + \overline z = z^2 + \frac{1}{z} (\because \ |z| = 1 \Rightarrow z \overline z=1)
= \sqrt z \left(z^{\frac{3}{2}} + \frac{1}{z^{\frac{3}{2}}} \right)
But if |z| =1, \frac{1}{z^{\frac{3}{2}}} = \overline {z^{\frac{3}{2}}} \Rightarrow z^{\frac{3}{2}} + \frac{1}{z^{\frac{3}{2}}} is a real number.
If further we have the required constraints on arg z, then it is a positive real number.
Hence, then \arg (S) = \arg \sqrt z= \frac{1}{2} \arg z
3412) |z| = 1 \Rightarrow z =\cos \theta + i \sin \theta
Let t = \tan \frac{\theta}{2}
Then z = \frac{1-t^2}{1+t^2} + \frac{2t}{1+t^2} i = \frac{1-t^2+2ti}{1+t^2} = \frac{(1-ti)^2}{(1+ti)(1-ti)}
=\frac{1-ti}{1+ti}
Setting c = \frac{1}{t} gives us the required result
1thanks all of u for ur posts. by the way i got my answers himself but in a different way.