7.16.
The Figure Effect of = z 2 on Various Domains.
obtained by cutting the complex plane along the negative real axis. On the other hand, vertical lines Re z = a are mapped to circles | | = ea . Thus, a vertical strip a < Re z < b is mapped to an annulus ea < | | < eb , albeit many-toone, since the strip is effectively wrapped around and around the annulus. The rectangle R = a < x < b, - < y < of height 2 is mapped in a one-to-one fashion on an annulus that has been cut along the negative real axis. See Figure 7.15. Finally, we note that no domain is mapped to the unit disk D = { | | < 1 } (or, indeed, any other domain that contains 0) because the exponential function is never zero: = ez = 0. Example 7.23. The squaring map = g(z) = z 2 , or = x2 - y 2 , = 2 x y, (7.56)
while the horizontal strip S- , = - < Im z < of width 2 is mapped onto the domain = - , = - < ph < = C \ { Im z = 0, Re z 0 }
is analytic on all of C, but is not one-to-one. Its inverse is the square root function z = , which, as we noted in Section 7.1, is doubly-valued, except at the origin z = 0. Furthermore, its derivative g (z) = 2 z vanishes at z = 0, violating the invertibility condition (7.49). However, once we restrict g(z) to a simply connected subdomain that does not contain 0, the function g(z) = z 2 does define a one-to-one mapping, whose inverse z = g -1 () = is a well-defined, analytic and single-valued branch of the square root function. The effect of the squaring map on a point z is to square its modulus, | | = | z |2 , while doubling its phase, ph = ph z 2 = 2 ph z. Thus, for example, the upper right quadrant Q= x > 0, y > 0 = 0 < ph z <
1 2
is mapped onto the upper half plane U = g(Q) = = Im > 0 = 0 < ph < .
The inverse function maps a point U back to its unique square root z = in the quadrant Q. Similarly, a quarter disk Q = 0 < | z | < , 0 < ph z <
1 2
that lies
of radius is mapped to a half disk U2 = g() = 0 < | | < 2 , Im > 0
of radius 2 . On the other hand, the unit square S = 0 < x < 1, 0 < y < 1 is mapped to a curvilinear triangular domain, as indicated in Figure 7.16; the edges of the square on 1/19/12 243
c 2012 Peter J. Olver
the real and imaginary axes map to the two halves of the straight base of the triangle, while the other two edges become its curved sides. Example 7.24. A particularly important example is the analytic map = x2 + y 2 - 1 2y z-1 = + i , z+1 (x + 1)2 + y 2 (x + 1)2 + y 2 (7.57)
where we established the formulae for its real and imaginary parts in (7.11). The map is one-to-one with analytic inverse z= 1+ 1 - 2 - 2 2 = + i , 2 + 2 1- (1 - ) (1 - )2 + 2 (7.58)
provided z = -1 and = 1. This particular analytic map has the important property of mapping the right half plane R = x = Re z > 0 to the unit disk D = | |2 < 1 . Indeed, by (7.58) | |2 = 2 + 2 < 1 if and only if x= 1 - 2 - 2 > 0. (1 - )2 + 2
Note that the does denominator not vanish on the interior of the disk D.
The complex functions (7.53, 54, 57) are particular examples of linear fractional transformations z + , (7.59) = z + which form one of the most important classes of analytic maps. Here , , , are complex constants, subject only to the restriction - = 0, since otherwise (7.59) reduces to a trivial constant (and non-invertible) map. (Why?) Example 7.25. The linear fractional transformation = z- , z -1 with | | < 1, (7.60)
Subtracting these two formulae,
maps the unit disk to itself, moving the origin z = 0 to the point = . To prove this, we note that | z - |2 = (z - )( z - ) = | z |2 - z - z + | |2 , | z - 1 |2 = ( z - 1)( z - 1) = | |2 | z |2 - z - z + 1. | z - |2 - | z - 1 |2 = 1 - | |2 |z -| <1 |z - 1| | z |2 - 1 < 0, whenever | z | < 1, | | < 1.
Thus, | z - | < | z - 1 |, which implies that | | = 1/19/12 provided 244 | z | < 1, | | < 1,
c 2012 Peter J. Olver
f
Figure 7.17.
A Conformal Map.
and hence, as promised, lies within the unit disk. The rotations (7.52) also map the unit disk to itself, while leaving the origin fixed. It can be proved, [3, 118], that the only invertible analytic mappings that take the unit disk to itself are obtained by composing such a linear fractional transformation with a rotation. Proposition 7.26. If = g(z) is a one-to-one analytic map that takes the unit disk to itself, then g(z) = e i z- z - 1 for some | | < 1, - < . (7.61)
Additional properties of linear fractional transformations are outlined in the exercises. Conformality A remarkable geometrical property enjoyed by all complex analytic functions is that, at non-critical points, they preserve angles, and therefore define conformal mappings. Conformality makes sense for any inner product space, although in practice one usually deals with Euclidean space equipped with the standard dot product. Definition 7.27. A function g: R n R n is called conformal if it preserves angles. But what does it mean to "preserve angles"? In the Euclidean norm, the angle between two vectors is defined by their dot product. However, most analytic maps are nonlinear, and so will not map vectors to vectors since they will typically map straight lines to curves. However, if we interpret "angle" to mean the angle between two curves , as illustrated in Figure 7.17, then we can make sense of the conformality requirement. Thus, in order to realize complex functions as conformal maps, we first need to understand their effect on curves. In general, a curve C C in the complex plane is parametrized by a complex-valued function z(t) = x(t) + i y(t), a t b, (7.62) that depends on a real parameter t. Note that there is no essential difference between a complex plane curve (7.62) and a real plane curve; we have merely switched from vector
Or, more precisely, the angle between their tangent vectors at the point of intersection; see below for details.
1/19/12
245
c 2012
Peter J. Olver