Page 430 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 421

1 7 9 9ðz 1Þ
þ ; double pole, 0 < jz 1j < 4
2 64 256
ðcÞ þ þ
16ðz 1Þ
4ðz 1Þ
RESIDUES AND THE RESIDUE THEOREM
16.63. Determine the residues of each function at its poles:
2z þ 3 z 3 e zt z
z 4 z þ 5z 2
ðaÞ 2 ; ðbÞ 3 2 ; ðcÞ 3 ; ðdÞ 2 :
ðz 2Þ ðz þ 1Þ
1 2 2t
Ans. (a) z ¼ 2; 7=4; z ¼ 2; 1=4 (c) z ¼ 2; t e
2
(b) z ¼ 0; 8=25; z ¼ 5; 8=25 (d) z ¼ i; 0; z ¼ i; 0
zt
16.64. Find the residue of e tan z at the simple pole z ¼ 3 =2.
Ans: e 3 t=2
z dz
þ 2
16.65. Evaluate , where C is a simple closed curve enclosing all the poles.
C ðz þ 1Þðz þ 3Þ
Ans: 8 i

16.66. If C is a simple closed curve enclosing z ¼ i,show that
ze 1
þ zt
dz ¼ t sin t
2 2 2
C ðz þ 1Þ
16.67. If f ðzÞ¼ PðzÞ=QðzÞ, where PðzÞ and QðzÞ are polynomials such that the degree of PðzÞ is at least two less than
þ
the degree of QðzÞ,prove that f ðzÞ dz ¼ 0, where C encloses all the poles of f ðzÞ.
C
EVALUATION OF DEFINITE INTEGRALS
Use contour integration to verify each of the following
ð 2 ð 2 p ffiffiffi
1 x dx d 4 3
16.68. ¼ p ffiffiffi 16.75. ¼
4
0 x þ 1 2 2 0 ð2 þ cos Þ 2 9
ð ð 2
dx 2 sin
1
16.69. 6 6 ¼ 5 ; a > 0 16.76. d ¼
1 x þ a 3a 0 5 4cos 8
dx d 3
ð ð 2
1
16.70. 16.77.
2 2 ¼ 32 2 2 ¼ p ffiffiffi
2 2
0 ðx þ 4Þ 0 ð1 þ sin Þ
cos n d
ð 2
ð p ffiffiffi 2 ¼
1 x 0 1 2a cos þ a
16.71. dx ¼ 16.78.
3
0 x þ 1 3 2 a n
1 a 2 ; n ¼ 0; 1; 2; 3; .. . ; 0 < a < 1
ð ð 2 2 2
1 dx 3 d ð2a þ b Þ
7
16.72. ¼ p a ; a > 0 16.79. ;
ffiffiffi
4 4 2 8 2 3 ¼ 2 2 5=2 a > jbj
0 ðx þ a Þ 0 ða þ b cos Þ ða b Þ
ð ð 4
dx x sin 2x e
1 1
16.73. 16.80.
2 2 2 ¼ 9 2 dx ¼ 4
1 ðx þ 1Þ ðx þ 4Þ 0 x þ 4
ð 2 d 2 ð 1 cos 2 x e
16.74. ¼ p ffiffiffi 16.81. dx ¼
0 2 cos 3 0 x þ 4 8
4