Page 747 - Advanced_Engineering_Mathematics o'neil

21.2 The Laurent Expansion 727

EXAMPLE 21.9
Let
1
f (z) = .
(z + 1)(z − 3i)
Then f is differentiable except at −1 and 3i. We will find the Laurent expansion of f (z) about
−1. Use a partial fractions decomposition to write
−1 + 3i 1 1 − 3i 1
f (z) = + .
10 z + 1 10 z − 3i
On the right, the first term is itself a Laurent expansion about −1, because it is a sum (with only
one term) of powers of z + 1. Therefore, concentrate on the second term. We will manipulate it
and use a geometric series, keeping in mind that we want a series of powers of z + 1:
1 1 1 1
= =
z − 3i −1 − 3i + (z + 1) −1 − 3i 1 − z+1
1+3i
n ∞
1 z + 1 −1 n
∞
=− = (z + 1) .
1 + 3i 1 + 3i (1 + 3i) n+1
n=0 n=0
This expansion is valid for

z + 1
< 1
1 + 3i

√
or |z + 1| < 10. The Laurent expansion of f (z) about −1is
∞ n+1
−1 + 3i 1 1 − 3i −1
f (z) = − (z + 1) n
10 z + 1 10 1 + 3i
n=0
√
in the annulus 0 < |z + 1| < 10. Behavior of f (z) as z approaches −1 is determined by the
1/(z + 1) term in this expansion.
We have emphasized that we do not want to have to use the integral formula for the c n ’s to
compute a Laurent expansion. This is really just the tip of the iceberg. Usually we are interested
in just one term of a Laurent expansion because it will enable us to evaluate integrals. This is the
theme of the next chapter.

SECTION 21.2 PROBLEMS

z 2
2
In each of Problems 1 through 10, write the Laurent expan- 7. e /z ;0
sion of f (z) in an annulus 0 < |z − z 0 | < R about z 0 , 8. sin(4z)/z;0
specifying R for each problem. These should all be done
9. (z + i)/(z − i);i
by manipulating known series.
3
10. sinh(1/z );0
11. Fill in the details of the following proof of the Laurent
2
1. 2z/(1 + z );i
expansion theorem (21.5). Let z be in the annulus, and
2
2. sin(z)/z ;0 choose r 1 and r 2 such that
2
3. (1 − cos(2z))/z ;0
2
4. z cos(i/z);0 0 <r <r 1 <r 2 < R
2
5. z /(1 − z);1 and so that the circle γ 1 :|z − z 0 |=r 1 does not enclose
2
6. (z + 1)/(2z − 1);1/2 z while the circle γ 2 :|z − z 0 |= r 2 encloses z.Insert
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October 14, 2010 15:35 THM/NEIL Page-727 27410_21_ch21_p715-728