(a) Find a harmonic function u(z, v) on the upper half-plane H := {(1, v) | v2 0} such that u is bounded and (0, lim u(z, y) = z>1 or z< -1 1, 0
(a) Find a harmonic function u(z, v) on the upper half-plane H := {(1, v) | v2 0} such that u is bounded and (0, lim u(z, y) = z>1 or z< -1 1, 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
This is related to dirichlet and harmonic functions.
![(a) Find a harmonic function u(r, y) on the upper half-plane H :=
{(r, y) | y2 0} such that u is bounded and
[0,
lim u(z, y) =
I>1 or z<-1
1,
0<r<1
-1, -1 <z < 0.
(b) Find the harmonic conjugate v(z, y) of u(r, y). Is it bounded?
(c) Find a harmonic function on the strip S = {(z, y) | 0 < y <} such
that
0, z>0
lim f(z, y) =
(1, z<0
and
lim f(r, y) =
-1,
I<0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7ca93b0-e148-4cc0-a287-0ce883e8b210%2F1d466b33-d755-44ad-a3ac-5b291857999a%2Fascf3i_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Find a harmonic function u(r, y) on the upper half-plane H :=
{(r, y) | y2 0} such that u is bounded and
[0,
lim u(z, y) =
I>1 or z<-1
1,
0<r<1
-1, -1 <z < 0.
(b) Find the harmonic conjugate v(z, y) of u(r, y). Is it bounded?
(c) Find a harmonic function on the strip S = {(z, y) | 0 < y <} such
that
0, z>0
lim f(z, y) =
(1, z<0
and
lim f(r, y) =
-1,
I<0.
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