Page 55 - Calculus with Complex Numbers

y == circle centre p, radius 2. 1n ttés case b0th singularities qzi are ùlside
y. lrherefore

y
j zl 2 - lx i ( sl - sl ) - O .

1
+
z

y == circle centre p, radius 1/2. 1n this case neither singularity is inside y.
lrherefore
dz
= O
r . z2 + 1
by Cauchy's theorem.

4 .8 Q u i c k ways of fi n d i ng resi d u es
For simple poles there are quicker methods for linding residues th= calculating
the Laurent expansion and taking the - 101 Laurent coeflicient. For example we
have the following.

Cover r!// rule lf flz) takes the form
fl g (z)
z) = ,
C - C

Proof The Taylor expansion for g (z) at z = c is

which gives immediately

fl g (c) ,
z) = + .!l (c) + ' ' '
C - C

Example Corlsider again
1 1
z = :
z q- 1 (z q- i ) (z -- i )
which has simple poles at z = Ljzi . Covering up z - i, z + i in turn we have

1 1 1 1 1 1
Rej = = , Res = = - .
k; c + i k; è-/ k; c - i k; è/
- -J cz + 1
-, .z2 + 1
- - g
= j