May 2024
Intermediate to advanced
481 pages
17h 15m
English
Content preview from Complex Analysis
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“ca: ch_09” — 2012/3/27 — 21:35 — page 443 — #27
Special Topics
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443
10. Show that the product
∞
n=2
1 −
1 −
1
n
−n
z
−n
converges absolutely for all z situated outside
the circle of radius unity and whose centre is at origin.
11. Show that the product
1 −
z
2
1
1 −
z
2
2
2
1 −
z
2
3
2
... converges absolutely and uni-
formly in the domain which does not contain the points ±1, ±2, ±3, ... whereas the product
1 −
z
1
1 +
z
1
1 −
z
2
1 +
z
2
... converges uniformly but not absolutely in the same
domain.
12. Show that the product
∞
n=1
1 +
1
n
z
is uniformly convergent in the domain Re (z)>1 +δ, where
δ is a positive number.
13. Show that
∞
n=1
1 +
1
n
sin
z
n
(a) converges absolutely ...
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