Liouville’s theorem: Suppose f ( z ) is analytic for all z (except ∞ ), and bounded [that is, | f ( z ) | ≤ M for all z and some M ] . Prove that f ( z ) is a constant. Hints: If f ′ ( z ) = 0 , then f ( z ) = const. To show this, write f ′ ( z ) as in Problem 3.21 where C is a circle of radius R and center z, that is, w = z + R e i θ . Show that f ′ ( z ) ≤ M / R and let R → ∞ .
Liouville’s theorem: Suppose f ( z ) is analytic for all z (except ∞ ), and bounded [that is, | f ( z ) | ≤ M for all z and some M ] . Prove that f ( z ) is a constant. Hints: If f ′ ( z ) = 0 , then f ( z ) = const. To show this, write f ′ ( z ) as in Problem 3.21 where C is a circle of radius R and center z, that is, w = z + R e i θ . Show that f ′ ( z ) ≤ M / R and let R → ∞ .
Liouville’s theorem: Suppose
f
(
z
)
is analytic for all
z
(except
∞
), and bounded [that is,
|
f
(
z
)
|
≤
M
for all
z
and some
M
]
.
Prove that
f
(
z
)
is a constant. Hints: If
f
′
(
z
)
=
0
,
then
f
(
z
)
=
const. To show this, write
f
′
(
z
)
as in Problem 3.21 where
C
is a circle of radius
R
and center z, that is,
w
=
z
+
R
e
i
θ
.
Show that
f
′
(
z
)
≤
M
/
R
and let
R
→
∞
.
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
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