Page 429 - Schaum's Outline of Theory and Problems of Advanced Calculus

420 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

ð
2
16.53. Evaluate ðz z þ 2Þ dz, where C is the upper half of the circle jzj¼ 1 tranversed in the positive sense.
C
Ans: 14=3
z;
þ
16.54. Evaluate , where C is the circle (a) jzj¼ 2; ðbÞjz 3j¼ 2:
C 2z 5
Ans: ðaÞ 0; ðbÞ 5 i=2

z
þ 2
16.55. Evaluate dz, where C is: (a)a square with vertices at 1 i; 1 þ i; 3 þ i; 3 i;
C ðz þ 2Þðz 1Þ
p ffiffiffi
(b)the circle jz þ ij¼ 3; (c)the circle jzj¼ 2.
Ans: ðaÞ 8 i=3 ðbÞ 2 i ðcÞ 2 i=3
cos z e þ z
þ þ z
16.56. Evaluate (a) dz; ðbÞ dz where C is any simple closed curve enclosing z ¼ 1.
C z 1 C ðz 1Þ 4
Ans: ðaÞ 2 i ðbÞ ie=3
16.57. Prove Cauchy’s integral formulas.
[Hint: Use the definition of derivative and then apply mathematical induction.]

SERIES AND SINGULARITIES
16.58. For what values of z does each series converge?
n n
1 1 1
X X X
; ; ð 1Þ ðz þ 2z þ 2Þ :
ðz þ 2Þ nðz iÞ n 2 2n
n! n þ 1
ðaÞ ðbÞ ðcÞ
n¼1 n¼1 n¼1
Ans: ðaÞ all z (b) jz ij < 1 ðcÞ z ¼ 1 i
z
1 n
X
16.59. Prove that the series is (a) absolutely convergent, (b) uniformly convergent for jzj @ 1.
n¼1 nðn þ 1Þ
n
1
X ðz þ iÞ
16.60. Prove that the series converges uniformly within any circle of radius R such that jz þ ij < R < 2.
2 n
n¼0
16.61. Locate in the finite z plane all the singularities, if any, of each function and name them:
2
z 2 z z þ 1 1 sinðz =3Þ cos z
4 2 z þ 2z þ 2 z 3z 2 2
2
ðaÞ ; ðbÞ ; ðcÞ ; ðdÞ cos ; ðeÞ ; ð f Þ :
ð2z þ 1Þ ðz 1Þðz þ 2Þ ðz þ 4Þ
1
Ans. (a) z ¼ , pole of order 4 (d) z ¼ 0, essential singularity
2
(b) z ¼ 1, simple pole; z ¼ 2, double pole (e) z ¼ =3, removable singularity
(c) simple poles z ¼ 1 i ( f ) z ¼ 2i, double poles
16.62. Find Laurent series about the indicated singularity for each of the following functions, naming the singu-
larity in each case. Indicate the region of convergence of each series.
cos z 2 1=z z 2
; z ¼ ðbÞ z e ; z ¼ 0 ; z ¼ 1
z
ðaÞ ðcÞ 2
ðz 1Þ ðz þ 3Þ
1 z 3 5
Ans: ðaÞ þ ðz Þ þ ðz Þ ; simple pole, all z 6¼
z 2! 4! 6!
1 1 1 1
2
2! 3! z 4! z 5! z
ðbÞ z z þ þ 2 3 þ ; essential singularity, all z 6¼ 0