Math 31BH  Homework 3. Due Wednesday, February 2.
Part I.
1.
(
Wednesday, January 26.
)
Laplacian and harmonic functions.
(i) The temperature
T
(
x, y
)
in a long thin plane at the point
(
x, y
)
satisfies Laplace’s equation
T
xx
+
T
yy
= 0
.
Does the function
T
(
x, y
) = ln(
x
2
+
y
2
)
satisfy Laplace’s equation?
(ii) Functions which satisfy the Laplace equation are called harmonic. Show that
T
(
x, y
) =
e
x
cos
y
is harmonic. See also Problem 5 (ii).
(iii) * The function in (i) has the special form
T
(
x, y
) =
h
(
x
2
+
y
2
)
where
h
(
z
) = ln
z
. Find all harmonic functions of the form
T
(
x, y
) =
h
(
x
2
+
y
2
)
.
(iv) The Laplacian of a function
f
(
x
1
, . . . , x
n
)
of
n
variables is defined to be
Δ
f
=
∂
2
f
∂x
2
1
+
. . .
+
∂
2
f
∂x
2
n
.
How would you define a harmonic function of
n
variables?
2.
(
Wednesday, January 26.
)
d’Alambertian and waves.
The d’Alambertian of the function
u
(
x
1
, . . . , x
n
, t
)
of
(
n
+ 1)
variables is defined to be
u
=
∂
2
u
∂x
2
1
+
. . .
+
∂
2
u
∂x
2
n

1
c
2
∂
2
u
∂t
2
.
This differs from the Laplacian defined in the previous problem by the choice of signs
(+
,
+
, . . . ,
+
,

)
in front of the derivatives. This choice of signs is dictated by special relativity. The constant
c
has
the physical interpretation as the speed.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '11
 DragosOprea
 Math, Taylor Series, Laplace, Analytic function, Holomorphic function

Click to edit the document details