Complex Analysis Answers

(a) Show that, for any complex number z, zz = |z|
2
, z + z = 2Re(z) and Re(z) ≤ |z|. Hence
show that
i. |z1 + z2|
2 = |z1|
2 + |z2|
2 + 2Re(z1z2),
ii. |z1 + z2| ≤ |z1| + |z2|,
where Re(z) is the real part of z and z the conjugate of z.

Show that, for any complex number z, zz = |z|
2
, z + z = 2Re(z) and Re(z) ≤ |z|. Hence
show that
i. |z1 + z2|
2 = |z1|
2 + |z2|
2 + 2Re(z1z2),
ii. |z1 + z2| ≤ |z1| + |z2|,
where Re(z) is the real part of z and z the conjugate of z. [26 marks]
(b) If z1 = 1 + 2i, find the set of values of z2 for which
(i) |z1 + z2| = |z1| + |z2| (ii) |z1 + z2| = |z1| − |z2|.

z^3=6 ( cos ( π/3 ) + i sin ( π/63 ) )

Obtain the 6th root of (-7)

Suppose that f(z) is analytic/holomorphic in Ω, an open connected set and |f(z)| < 1 for z ∈ Ω. Show that the function defined by g(z) = Summation from n=1 to infinity n{f(z)}^n is also holomorphic in Ω

im(z+9/z)=0

The transformation T : z "→" w in the complex plane is defined by w = (az + b)/( z + c), where a, b, c ∈ R. Given that w = 3i when z = −3i, and w = 10 − 4i when z = 1 + 4i, find the values of a, b and c. (a) Show that the points for which z is transformed to z lie on a circle and give the centre and radius of this circle. (b) Show that the line through the point z = 4 and perpendicular to the real axis is invariant under T.

The transformation T : z "→" w in the complex plane is defined by w = (az + b)/( z + c), where a, b, c ∈ R. Given that w = 3i when z = −3i, and w = 10 − 4i when z = 1 + 4i, find the values of a, b and c. (a) Show that the points for which z is transformed to z lie on a circle and give the centre and radius of this circle. (b) Show that the line through the point z = 4 and perpendicular to the real axis is invariant under T.

The transformation T : z "→" w in the complex plane is defined by w = (az + b)/( z + c), where a, b, c ∈ R. Given that w = 3i when z = −3i, and w = 10 − 4i when z = 1 + 4i, find the values of a, b and c. (a) Show that the points for which z is transformed to z lie on a circle and give the centre and radius of this circle. (b) Show that the line through the point z = 4 and perpendicular to the real axis is invariant under T.

Find the Cartesian equation of the locus of the point P representing the complex
number z. Sketch the locus of P in each case.
(a) 2|z + 1| = |z − 2|