204 Functions of a Complex VariableEXERCISE 4.2.24. B Using the Cauchy-Riemann equations (4-2.2), verify
that the Jacobian of the mapping defined by an analytic function f = f(z)
is equal to \f'(z)\^2.
Beside the conservation of angles, another important property of the
mapping defined by an analytic function with non-zero derivative is the
uniform scaling at every point, that is, the linear dimensions near every
point z are changed by the same factor |/'(z)| near w = f{z) in all direc-
tions. This property is the consequence of the definition of the derivative:
if wo = f(zo), u> = f(z), Az = z~ ZQ, AW = w — wo, then, for \Az\ close to
zero, we have (Aw/Az) « f'(zo) or \Aw\ « |/'(^o)| |Az|, and the direction
from ZQ to z or from WQ to w does not matter.
By definition, a mapping is called conformal at a point if it preserves
angles and has uniform scaling at that point. We just saw that an analytic
function defines a conformal mapping at all points where the derivative of
the function is non-zero.
EXERCISE 4.2.25.B Let f = f(z) be an analytic function in a domain G
and f'{z) ^ 0 in G. (a) Denoting by G the image of G under f, verify that
the area of G is Jf \f'(z)\^2 dA. (b) Denoting by C the image under f of a
G
piece-wise smooth curve C in G, show that the length of C is J \f'(z)\ \dz.
c
Figuring out how a particular function / transforms a certain domain or
a curve is usually a matter of straightforward computations. In doing these
calculations, one should keep in mind that, while a conformal mapping pre-
serves the local geometry, the global geometry can change quite dramatically.
FOR EXAMPLE, let us see what the mapping f(z) = 1/z, which is confor-
mal everywhere except z = 0, does to the family of circles \z — ic\^2 = c^2 ,
where c > 0 is a real number. It is convenient to consider two different
complex planes: where the function / is defined, and where the function /
takes its values. Since z = x + iy denotes the generic complex number in
the complex plane where / is denned, it is convenient to introduce a differ-
ent letter, w, to denote the generic complex number in the complex plane
where / takes its values. The equation of the circle is x^2 + y^2 — 2cy = 0.
The relation between z and w is w = 1/z = (x — iy)/(x^2 + y^2 ). When z
is on the circle, we have x^2 + y^2 = 2cy, and then w = x/(2cy) — i/(2c).
In other words, if a point z is on the circle \z — ic\^2 = c^2 , then the point
w = 1/z satisfies Sw = —l/(2c); you should convince yourself that, as we
take different points z on the circle, we can get all possible values of the