Page 427 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 427
418 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16
z
þ p 1
Consider dz.Since z ¼ 0isa branch point, choose C as the contour of Fig. 16-14 where AB
C 1 þ z
and GH are actually coincident with x-axis but are shown separated for visual purposes.
The integrand has the pole z ¼ 1lying within C.
Residue at z ¼ 1 ¼ e i is
z p 1 i p 1 ðp 1Þ i
lim ðz þ 1Þ ¼ðe Þ ¼ e
z! 1 1 þ z
þ p 1
z
Then dz ¼ 2 ie ðp 1Þ i
C 1 þ z
or, omitting the integrand,
ð ð ð ð
¼ 2 ie ðp 1Þ i
þ þ þ
AB BDEFG GH HJA
We thus have
x ðRe Þ iRe d ðxe Þ dx
ð R p 1 ð 2 i p 1 i ð r 2 i p 1
r 1 þ x dx þ 0 1 þ Re i þ R 1 þ xe 2 i
i
ð 0 i p 1 ire d Fig. 16-14
ðre Þ ðp 1Þ i
þ i ¼ 2 ie
2 1 þ re
where we have to use z ¼ xe 2 i for the integral along GH, since the argument of z is increased by 2 in going
round the circle BDEFG.
Taking the limit as r ! 0 and R !1 and noting that the second and fourth integrals approach zero,
we find
ð p 1 ð 0 2 iðp 1Þ p 1
x e x
1
dx ¼ 2 e ðp 1Þ i
0 1 þ x 1 1 þ x
dx þ
ð p 1
1 x
or ð1 e 2 iðp 1Þ Þ dx ¼ 2 ie ðp 1Þ i
0 1 þ x
so that
ð p 1 ðp 1Þ i
1 x 2 ie 2 i
0 1 þ x dx ¼ 1 e 2 iðp 1Þ ¼ e p i e p i ¼ sin p
Supplementary Problems
FUNCTIONS, LIMITS, CONTINUITY
16.39. Describe the locus represented by (a) jz þ 2 3ij¼ 5; ðbÞjz þ 2j¼ 2jz 1j; ðcÞjz þ 5j jz 5j¼ 6.
Construct a figure in each case.
2
2
Ans. ðaÞ Circle ðx þ 2Þ þð y 3Þ ¼ 25, center ð 2; 3Þ,radius 5.
2
2
(b)Circle ðx 2Þ þ y ¼ 4, center ð2; 0Þ,radius 2.
2
2
(c)Branch of hyperbola x =9 y =16 ¼ 1, where x A 3.
16.40. Determine the region in the z plane represented by each of the following:
(a) jz 2 þ ij A 4; ðbÞjzj @ 3; 0 @ arg z @ ; ðcÞjz 3jþjz þ 3j < 10.
4
Construct a figure in each case.
2
2
Ans.(a) Boundary and exterior of circle ðx 2Þ þð y þ 1Þ ¼ 16.