1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 393

Equations ( 6.24-6) are identical (mu ta tis mutandi) to the Beltrami

equations (6.24-2). For we may let, assuming that 1771 < 1,

E = 17712 +2a+1, F = 2/3

H = JEG-F^2 = 1-1771^2 (6.24-7)

We note that equation ( 6.24-4) yields

d0'^2 = ldwl^2 = lfzdz + fzdzl^2 = lfzl^2 ldz + 71dzl^2

so that the mapping defined by w = f (z) is conformal, with magnification

ratio p = lfzl, if we adopt on the z-plane the Riemannian metric

ds^2 = ldz + 71dzl^2

= (1771^2 + 2a + l)dx^2 + 4f3dxdy + (1771^2 - 2a + l)dy^2

= Edx^2 + 2Fdxdy + Gdy^2

With the assumption J1(z) > O, a K-quasiconformal mapping f: R---+

f(R) is defined as a mapping for which the dilatation D 1 is bounded over

R, i.e, such that

Dt = lfzl + lfzl = 1+1771 < K

lfzl - lfzl 1 - 1771 -

where K = lubzeR D f < oo, or, equivalently, for which the complex dilation

is such that

fz

(^77) =::; f z
K-1
l'Yll< ., - --K+l =k
Thus the discussion above gives the following remarkable result: A K-
quasiconformal mapping with positive Jacobian is equivalent to a conformal
mapping with respect to the Riemannian metric determined by the complex
dilatation 77, i.e, that with parameters (6.24-7). Hedrick and Ingold ([59
and [60]) have extended the Beltrami equations to three-dimensional space,
and J. D. Abercrombie [1] to four dimensions.
(b) In 1887 the Italian mathematician V. Volterra introduced the idea
of function of lines (the term functional, proposed by Hadamard, has now
replaced Volterra's terminology), and in the period 1887-1890 he made use
of the new concept to extend the notion of conjugate functions in the plane

to that between an ordinary function f in space (or hyperspace) and a

function of lines F[L]. In the three-dimensional case Volterra says that a
differentiable function f(x, y, z) and a function of lines F[L] are mutually