24.7 Conformal transformations
Thus limz→ 0 f(z) = 1 independently of the way in whichz→0, and sof(z) has a removable
singularity atz=0.
An expression common in mathematics, but which we have so far avoidedusing explicitly in this chapter, is ‘ztends to infinity’. For a real variable such
as|z|orR, ‘tending to infinity’ has a reasonably well defined meaning. For a
complex variable needing a two-dimensional plane to represent it, the meaning is
not intrinsically well defined. However, it is convenient to have a unique meaning
and this is provided by the followingdefinition: the behaviour off(z)at infinity
is given by that off(1/ξ)atξ=0,whereξ=1/z.
Find the behaviour at infinity of(i)f(z)=a+bz−^2 ,(ii)f(z)=z(1 +z^2 )and(iii)
f(z)=expz.(i)f(z)=a+bz−^2 : on puttingz=1/ξ,f(1/ξ)=a+bξ^2 , which is analytic atξ=0;
thusfis analytic atz=∞.
(ii)f(z)=z(1 +z^2 ):f(1/ξ)=1/ξ+1/ξ^3 ; thusfhas a pole of order 3 atz=∞.
(iii)f(z)=expz:f(1/ξ)=∑∞
0 (n!)− (^1) ξ−n; thusfhas an essential singularity atz=∞.
We conclude this section by briefly mentioning thezerosof a complex function.
As the name suggests, iff(z 0 )=0thenz=z 0 is called a zero of the function
f(z). Zeros are classified in a similar way to poles, in that if
f(z)=(z−z 0 )ng(z),
wherenis a positive integer andg(z 0 )=0,thenz=z 0 is called azero of order
noff(z). Ifn=1thenz=z 0 is called asimple zero. It may further be shown
that ifz=z 0 is a zero of ordernoff(z) then it is also a pole of ordernof the
function 1/f(z).
We will return in section 24.11 to the classification of zeros and poles in terms
of their series expansions.
24.7 Conformal transformations
We now turn our attention to the subject of transformations, by which we mean
a change of coordinates from the complex variablez=x+iyto another, say
w=r+is, by means of a prescribed formula:
w=g(z)=r(x, y)+is(x, y).
Under such a transformation, ormapping, the Argand diagram for thez-variable
is transformed into one for thew-variable, although the completez-plane might
be mapped onto only a part of thew-plane, or onto the whole of thew-plane, or
onto some or all of thew-plane covered more than once.
We shall consider only those mappings for whichwandzare related by a
functionw=g(z) and its inversez=h(w) with both functions analytic, except
possibly at a few isolated points; such mappings are calledconformal.Their