1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 265

multiplicity. At any such pole (3 we have P((J) f:. 0, Q((J) = O, so R((J) =

P((J) / Q((J) = oo. Hence R( z) has n zeros and m poles in C, counting

multiplicities.

The definition of w = R(z) is extended to the point oo by means of

the following.

Definition 5.2 Let R(l/z) = R 1 (z). Then we define

R(oo) = Ri(O)

Since

Ri(z) = ao + ai(l/z) + · · · + an(l/zn)

bo + b1(l/z) + · · · + bm(l/zm)

m-n aoz n·+ aiz n-1 + + · · · an

=z

bozm + b1zm-l + · · · + bm

if m > n, then R(oo) = R 1 (0) = 0, so in this case oo is a zero of R(z) of
order m - n; if m = n, then R( oo) = Ri (0) = an/bm, so oo is neither a
zero nor a pole of R(z); finally, if m < n, then R(oo) = R 1 (0) = oo, so in
this case oo is a pole of R( z) of order n - m.
Thus, counting the total number of zeros and poles of R( z) in C*, we have

Number Number

Case of zeros of poles

m>n n+(m-n)=m m

m=n n m

m<n n m+(n-m)=n

We see that in any case the number of zeros of R(z) in the extended

complex plane equals the number of its poles. It we denote that number

by p, we have p = max(m,n). The number pis called the order of the
rational function.

Examples

1. For the function w = (z^4 - 1)/(z^2 + 4) we have p = max(2,4) = 4, so
   it has in C* four zeros: 1, -1, i, -i, and four poles: 2i, -2i, oo, oo.

2. For w = (z^2 - 9)/(z^3 - 1), p = 3 so it has three zeros: 3, -3, oo and

three poles: 1, 'f], "7^2 ("7 = e^2 '1ri/^3 ).

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