264 Chapter^5
where 0 < B 1 - Bo :::; 27r /n, i.e., the interior of the angle formed by the rays
Lo and L 1 (Fig. 5.20). Then its image under w = (z -ar is the region
R' = {w: 0 < lwl < oo,nB 0 < argw < nBi}
i.e., the interior of the angle formed by the rays L~ and L~. Under the
assumption 0 < B 1 - Bo :::; 27r /n, it is easy to verify that the function
w = (z - a)n is one-to-one in R. Because w = (z - ar is single-valued,
it suffices to observe that every w E R' has just one inverse image in R.
In fact, since the n inverse images of w in the z-plane lie at the vertices of
a regular polygon of n sides with center at a, two of those inverse images
can lie in the interior of an angle with vertex at a iff the measure of that
angle is larger than 27r /n. But B 1 - B 0 :::; 27r /n, so that only one inverse
image o:f w belongs to the region R.
The concentric circles with center a and the rays issued from a are or-
thogonal to each other at every point of intersection and the same property
holds for their images at the corresponding points, the sense of the an-
gles being preserved, i.e., the mapping of these figures by w = (z - ar
is directly isogonal. The property of preserving angles in magnitude
and orientation is a general property of this function, as well as of all
other "analytic" functions, a property that fails only at certain excep-
tional points, for instance, at the points z = a and z = oo in the case
considered above. This property, usually known as the conformality prop-
erty is discussed in Section 6.15 and in more detail in Selected Topics,
Chapter 1.
5.16 The Rational Function w = P(z)/Q(z)
By a rational function it is understood the function defined by the quotient
of two polynomials of degrees n and m, respectively, namely,R( )
ao + a1z + · · · + anzn
w= z =
bo + b1z + · · · + bmzm(5.16-1)with an -=f. 0, bm -=f. 0. In what follows it will be assumed that the poly-
nomials P(z) and Q(z) are prime to each other, i.e., that they have nocommon factors o:f the form z - c. It should be noted that a polynomial
function (sometimes called a rational integral function) is a particular case
of a rational function since Q(z) can be taken to be a nonzero constant
polynomial.
The zeros, if any, of P( z) are the finite zeros of R( z ), since P( a) = O,
Q( a) -=f. 0 implies that R( a) = 0, and conversely. The order or multiplicity
of a zero of R(z) is the same as its order in P( z). On the other hand,
the zeros, if any, of Q(z) are called the finite poles of R(z), with the same