Complex Analysis Answers

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|

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.

Let x, y, u, v be real numbers and z, w be complex numbers such that z = x + iy, w = u+iv. In each case, describe the locus of the point (u, v) in the Argand diagram.

(a) w = z + 4 and (i) |z| = 3, (ii) arg z = π/3, (iii) |z + 4| = 5, (iv) y2 = 4x

In an Argand diagram, the point P represents the complex number z, where z = x+iy. Given that z+2 = λi(z+8), where λ is a real parameter, find the Cartesian equation of the locus of P as λ varies. If also z = µ(4 + 3i), where µ is real, prove that there is only one possible position for P.

Find two Laurent series in powers of z gor the function f defined by f(z)=1/z^2(1-z) and specify the regions in which the series converges to f(z).

Express −1+i in polar form. Hence show that (−1+i)16 is real and that 1/(−1+i)6 is purely imaginary, giving the value for each.