298 CHAPTER 8 • RESIDUE THEORY
which is valid for lz - al < le - al. Thus, the Laurent series off about the point
a is
A00
[ B C ] nf (z) = z - a -~ (b -a)"+^1 + (c - a)"+l (z -a) '
which is valid for lz - al < R, where R = min {lb-al, le -al}. Therefore, A =
Res[!, a}, and calculation reveals thatRes[Ja]=A=lim P(z) = P(a).
' z-a (z - b)(z - c) (a -b)(a - c)•EXAMPLE 8.8 Express/ (z) = z(z~f~;_ 2 > in partial fractions.
Solution Computing t he residues, we obtain
Res [j, OJ = 1, Res[!, 1 J = - 5, and Res [f , 2) = 4.
Example 8. 7 gives us3z + 2 1 5 4
-----= - ---+ --.
z(z-l)(z-2) z z-1 z-2Remark 8. 1 If a repeated root occurs, then the process is similar, and we can
easily show that if P (z) has degree of at most 2, then
f ( )
_ P (z) _ A B C
z - 2 - 2 + --+ --,
(z -a) (z - b) (z -a) z - a z - bwhe re A= Res[(z - a) f (z), a}, B =Res[/, a], and C = Res[f,b).
- EXAMPLE 8.9 Express f (z) = ~»t;.:t{ in partial fractions.
Solution Using the previous remark, we haveA B C
f(z)= 2 +--+-,
(z -a) z -a z - b