Page 416 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 407

n nþ1 n
1
X 3
: We have lim :
ðz iÞ u nþ1 ðz iÞ jz ij
3 n!1 u n n!1 3 ðz iÞ 3
ðcÞ n ¼ lim nþ1 n ¼

n¼1
The series converges if jz ij < 3, and diverges if jz ij > 3.
1
i X in
If jz ij¼ 3, then z i ¼ 3e and the series becomes e . This series diverges since the nth
n¼1
term does not approach zero as n !1.
Thus, the series converges within the circle jz ij¼ 3 but not on the boundary.
1
X n
16.19. If a n z is absolutely convergent for jzj @ R, show that it is uniformly convergent for these
n¼0
values of z.
The definitions, theorems, and proofs for series of complex numbers are analogous to those for real
series.
1
n
X
n
In this case we have ja n z j @ ja n jR ¼ M n . Since by hypothesis M n converges, it follows by the
n¼1
1
X
n
Weierstrass M test that a n z converges uniformly for jzj @ R.
n¼0
16.20. Locate in the finite z plane all the singularities, if any, of each function and name them.
z 2
: z ¼ 1is a pole of order 3.
3
ðaÞ
ðz þ 1Þ
3
2z z þ 1
(b) . z ¼ 4isa pole of order 2 (double pole); z ¼ i and z ¼ 1 2i are poles of
2
ðz 4Þ ðz iÞðz 1 þ 2iÞ
order 1 (simple poles).
ffiffiffiffiffiffiffiffiffiffiffi
p
sin mz 4 8 2 2i
2
(c) , m 6¼ 0. Since z þ 2z þ 2 ¼ 0 when z ¼ 2 ¼ ¼ 1 i,wecan write
2
z þ 2z þ 2 2 2
2
z þ 2z þ 2 ¼fz ð 1 þ iÞgfz ð 1 iÞg ¼ ðz þ 1 iÞðz þ 1 þ iÞ.
The function has the two simple poles: z ¼ 1 þ i and z ¼ 1 i.
1 cos z 1 cos z
(d) . z ¼ 0 appears to be a singularity. However, since lim ¼ 0, it is a removable
z x!0 z
singularity.
Another method:
( !)
1 cos z 1 z 2 z 4 z 6 z z 3
Since ¼ 1 1 þ þ ¼ þ ,we see that z ¼ 0isa remova-
z z 2! 4! 6! 2! 4!
ble singularity.
2 1 1
e 1=ðx 1Þ :
2 4
ðeÞ ¼ 1 þ
ðz 1Þ 2!ðz 1Þ
This is a Laurent series where the principal part has an infinite number of non-zero terms. Then
z ¼ 1isan essential singularity.
z
( f ) e .
This function has no finite singularity. However, letting z ¼ 1=u,we obtain e 1=u , which has an
z
essential singularity at u ¼ 0. We conclude that z ¼1 is an essential singularity of e .
In general, to determine the nature of a possible singularity of f ðzÞ at z ¼1,we let z ¼ 1=u and
then examine the behavior of the new function at u ¼ 0.
16.21. If f ðzÞ is analytic at all points inside and on a circle of radius R with center at a, and if a þ h is any
point inside C, prove Taylor’s theorem that