Show that a harmonic function u ( x , y ) is equal at every point a to its average value on any circle centered at a [and lying in the region where f ( z ) = u ( x , y ) + i v ( x , y ) is analytic). Hint: In ( 3.9 ) , let z = a + r e i θ (that is, C is a circle with center at a), and show that the average value of f ( z ) on the circle is f ( a ) (see Chapter 7, Section 4 for discussion of the average of a function). Take real and imaginary parts of f ( a ) = [ u ( x , y ) + i v ( x , y ) ] z = a .
Show that a harmonic function u ( x , y ) is equal at every point a to its average value on any circle centered at a [and lying in the region where f ( z ) = u ( x , y ) + i v ( x , y ) is analytic). Hint: In ( 3.9 ) , let z = a + r e i θ (that is, C is a circle with center at a), and show that the average value of f ( z ) on the circle is f ( a ) (see Chapter 7, Section 4 for discussion of the average of a function). Take real and imaginary parts of f ( a ) = [ u ( x , y ) + i v ( x , y ) ] z = a .
Show that a harmonic function
u
(
x
,
y
)
is equal at every point
a
to its average value on any circle centered at
a
[and lying in the region where
f
(
z
)
=
u
(
x
,
y
)
+
i
v
(
x
,
y
)
is analytic). Hint:
In
(
3.9
)
,
let
z
=
a
+
r
e
i
θ
(that is,
C
is a circle with center at a), and show that the average value of
f
(
z
)
on the circle is
f
(
a
)
(see Chapter 7, Section 4 for discussion of the average of a function). Take real and imaginary parts of
f
(
a
)
=
[
u
(
x
,
y
)
+
i
v
(
x
,
y
)
]
z
=
a
.
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
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