24.7 CONFORMAL TRANSFORMATIONS
point of intersection. Since any finite-length tangent may be curved,wiis more
strictly given bywi−w 0 =ρiexpi(φi+δφi), whereδφi→0asρi→0, i.e. as
ρ→0.
Now sincew=g(z), wheregis analytic, we havelim
z 1 →z 0(
w 1 −w 0
z 1 −z 0)
= lim
z 2 →z 0(
w 2 −w 0
z 2 −z 0)
=dg
dz∣
∣
∣
∣
z=z 0,which may be written as
lim
ρ→ 0{
ρ 1
ρexp[i(φ 1 +δφ 1 −θ 1 )]}
= lim
ρ→ 0{
ρ 2
ρexp[i(φ 2 +δφ 2 −θ 2 )]}
=g′(z 0 ).
(24.25)Comparing magnitudes and phases (i.e. arguments) in the equalities (24.25) gives
the stated results (ii) and (iii) and adds quantitative information to them, namely
that forsmallline elements
ρ 1
ρ≈ρ 2
ρ≈|g′(z 0 )|, (24.26)φ 1 −θ 1 ≈φ 2 −θ 2 ≈argg′(z 0 ). (24.27)For strict comparison with result (ii), (24.27) must be written asθ 1 −θ 2 =φ 1 −φ 2 ,
with an ordinary equality sign, since the angles are only defined in the limit
ρ→0 when (24.27) becomes a true identity. We also see from (24.26) that
the linear magnification factor is|g′(z 0 )|; similarly, small areas are magnified by
|g′(z 0 )|^2.
Since in the neighbourhoods of corresponding points in a transformation anglesare preserved and magnifications are independent of direction, it follows that small
plane figures are transformed into figures of the same shape, but, in general, ones
that are magnified and rotated (though not distorted). However, we also note
that at points whereg′(z) = 0, the angle argg′(z) through which line elements are
rotated is undefined; these are calledcritical pointsof the transformation.
The final result (iv) is perhaps the most important property of conformaltransformations. Iff(z) is an analytic function ofzandz=h(w) is also analytic,
thenF(w)=f(h(w)) is analytic inw. Its importance lies in the further conclusions
it allows us to draw from the fact that, sincefis analytic, the real and imaginary
parts off=φ+iψare necessarily solutions of
∂^2 φ
∂x^2+∂^2 φ
∂y^2= 0 and∂^2 ψ
∂x^2+∂^2 ψ
∂y^2=0. (24.28)Since the transformation property ensures thatF=Φ+iΨ is also analytic, we
can conclude that its real and imaginary parts must themselves satisfy Laplace’s
equation in thew-plane:
∂^2 Φ
∂r^2+∂^2 Φ
∂s^2= 0 and∂^2 Ψ
∂r^2+∂^2 Ψ
∂s^2=0. (24.29)