Transformations from the z-plane to the w-planeEdexcel A-Level Further Maths: Revision notes

A transformation $T$ given by $w=f(z)$ takes each point $z=x+iy$ on an Argand diagram (the $z$-plane) to a point $w=u+iv$ on a second Argand diagram (the $w$-plane). A locus in the $z$-plane, such as a line or circle, is mapped to a locus in the $w$-plane called its image. To find an image, write $x$ and $y$ in terms of $u$ and $v$ (or $z$ in terms of $w$), then substitute into the equation of the locus. Always finish with an equation in $u$ and $v$ only, and say what it describes: a line, circle or half-line.

For $w=az+b$ with $a,b$ complex, $z\to az$ is an enlargement by scale factor $|a|$ together with a rotation about $O$ through $\arg a$; adding $b$ is a translation. Circles map to circles and lines map to lines. Example: $w=3z+2i$. Then $z=\frac{w-2i}{3}$, so $x=\frac u3$ and $y=\frac{v-2}{3}$. The line $y=x$ becomes $\frac{v-2}{3}=\frac u3$, i.e. $v=u+2$. The circle $|z|=2$ becomes the circle with centre $2i$ and radius $6$.

Put $z=x+iy$: $w=x^2-y^2+2xyi$, so $u=x^2-y^2$ and $v=2xy$. In modulus-argument form, $z=re^{i\theta}$ maps to $w=r^2e^{2i\theta}$: the modulus is squared and the argument is doubled. So the circle $|z|=r$ maps to the circle $|w|=r^2$, and the half-line $\arg z=\alpha$ maps to the half-line $\arg w=2\alpha$. The line $y=x$ gives $u=0$, $v=2x^2\geq0$: only the non-negative imaginary axis. Because $z$ and $-z$ have the same square, the map is two-to-one.

Make $z$ the subject: from $w=\frac{az+b}{cz+d}$ we get $z=\frac{b-dw}{cw-a}$. Substitute into the equation of the locus, then simplify. Two methods are common:

• Cartesian: $z=\frac1w=\frac{u-iv}{u^2+v^2}$, so $x=\frac{u}{u^2+v^2}$ and $y=\frac{-v}{u^2+v^2}$.
• Modulus: for a locus such as $|z|=k$, use $|z_1z_2|=|z_1||z_2|$ and $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$ to reach an equation in $|w-\ldots|$ directly. For $w=\frac{z+i}{z-i}$, $z=\frac{i(w+1)}{w-1}$, so $|z|=3$ becomes $|w+1|=3|w-1|$, which gives a circle with centre $\left(\frac54,0\right)$ and radius $\frac34$.

For $w=\frac1z$: the circle $|z|=r$ maps to the circle $|w|=\frac1r$. The line $x=1$ gives $u^2+v^2=u$, a circle with centre $\left(\frac12,0\right)$ and radius $\frac12$. A circle through the point where the denominator is zero behaves differently: the circle $|z-1|=1$ passes through $z=0$, and $\left|\frac1w-1\right|=1$ gives $|1-w|=|w|$, i.e. the line $u=\frac12$. In general a Möbius map sends circles and lines to circles or lines; a circle through the point where the denominator is zero becomes a line.