Page 419 - Schaum's Outline of Theory and Problems of Advanced Calculus

410 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

þ
f ðzÞ dz ¼ 2 ia 1
ðaÞ
C n 1
1 d
n
ðbÞ a 1 ¼ lim fðz aÞ f ðzÞg:
z!a ðn 1Þ! dz n 1
(a)Byintegration, we have on using Problem 16.13
þ þ þ þ
a n a 1 2
f ðzÞ dz ¼ n dz þ þ dz þ fa 0 þ a 1 ðz aÞþ a 2 ðz aÞ þ g dz
C C ðz aÞ C z a C
¼ 2 ia 1
Since only the term involving a 1 remains, we call a 1 the residue of f ðzÞ at the pole z ¼ a.
n
(b) Multiplication by ðz aÞ gives the Taylor series
n n 1
ðz aÞ f ðzÞ¼ a n þ a nþ1 ðz aÞþ þ a 1 ðz aÞ þ
Taking the ðn 1Þst derivative of both sides and letting z ! a,we find
d n 1
ðn 1Þ!a 1 ¼ lim n
z!a dz n 1 fðz aÞ f ðzÞg
from which the required result follows.

16.24. Determine the residues of each function at the indicated poles.
z 2
2
ðaÞ ; z ¼ 2; i; i: These are simple poles. Then:
ðz 2Þðz þ 1Þ
( 2 )
z 4
Residue at z ¼ 2is lim ðz 2Þ ¼ :
z!2 2 5
ðz 2Þðz þ 1Þ
( 2 ) 2
z i 1 2i
Residue at z ¼ i is lim ðz iÞ ¼ ¼ :
z!i 10
ðz 2Þðz iÞðz þ iÞ ði 2Þð2iÞ
( 2 ) 2
z i 1 þ 2i
Residue at z ¼ i is lim ðz þ iÞ ¼ ¼ :
z! i ðz 2Þðz iÞðz þ iÞ ð i 2Þð 2iÞ 10
1
; z ¼ 0; 2: z ¼ 0isasimple pole, z ¼ 2isa pole of order 3. Then:
3
ðbÞ
zðz þ 2Þ
1 1
Residue at z ¼ 0is lim z ¼ :
z!0 3 8
zðz þ 2Þ
1 d 2 3 1
Residue at z ¼ 2is lim ðz þ 2Þ
z! 2 2! dz 2 3
zðz þ 2Þ
1 d 2 1 1
2
1
¼ lim ¼ lim ¼ :
z! 2 2 dz 2 z z! 2 2 z 3 8
Note that these residues can also be obtained from the coefficients of 1=z and 1=ðz þ 2Þ in the
respective Laurent series [see Problem 16.22(e)].
ze zt
ðcÞ 2 ; z ¼ 3; a pole of order 2 or double pole. Then:
ðz 3Þ
d ze zt d
Residue is lim 2 ¼ lim zt zt zt
z!3 dz ðz 3Þ 2 z!3 dz ðze Þ¼ lim ðe þ zte Þ
z!3
ðz 3Þ
3t
¼ e þ 3te 3t