Transformations from the z-plane to the w-planeEdexcel International A Level Further Maths: Flashcards

What these 13 flashcards ask

What do |w| and \arg w become under w=z^2?
Image of the circle |z|=r under w=z^2?
Image of the half-line \arg z=\alpha under w=z^2?
For w=z^2 with z=x+iy, what are u and v?
How is w=az+b described geometrically?
Image of |z-z0|=r under w=az+b?
How do you find z from w=\frac{az+b}{cz+d}?
What is the pole of w=\frac{az+b}{cz+d}?
A line or circle through the pole under a Möbius map gives what?
A circle not through the pole under a Möbius map gives what?
A line not through the pole under a Möbius map gives what?
Image of the circle |z|=r under w=\frac1z?
How do you find the image of a region?

Exam questions on Transformations from the z-plane to the w-plane

The transformation $T$ from the $z$ -plane to the $w$ -plane is given by $w=z^2$ , where $z=x+iy$ and $w=u+iv$ .
Find a Cartesian equation for the image of the line $x=2$ under $T$ .2 marks
The transformation $T$ from the $z$ -plane to the $w$ -plane is given by $w=(1+i)z-2$ .
The circle $|z|=1$ is mapped by $T$ to a circle in the $w$ -plane. Find its centre and radius.2 marks
The transformation $T$ from the $z$ -plane to the $w$ -plane is given by $w=\frac{z+i}{z-i}$ , $z\neq i$ , where $z=x+iy$ and $w=u+iv$ .
Show that the image of the real axis ( $y=0$ ) lies on the circle $|w|=1$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).