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6.9 Dirichlet Problem for the Upper Half-Plane
y
x
x-xfxu
yx-yxu
for)()0,(
0 ,for0),(Consider the following problem:
d
xy
fyyxu 22 )(
)(),(
We solved this problem by Fourier Transform:
We will solve this problem again by Fourier Integral: Goal
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6.9 Dirichlet Problem for the Upper Half-Plane
x-xfxu
yx-yxu
for)()0,(
0 ,for0),(Consider the following problem:
We will solve this problem again by Fourier Integral:
We looking for a bounded solution.
Goal
The Fourier Integral representation of f
0
)sin()cos( dwwxBwxA ww
dwfAw )cos()(
1
dwfBw )sin()(
1
Fourier integral coefficients of f
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6.9 Dirichlet Problem for the Upper Half-Plane
x-xfxu
yx-yxu
for)()0,(
0 ,for0),(Consider the following problem: )()(),( yYxXyxu
Y
Y
X
X ''''
0 λYY''0'' XX constant :0 X
sol trivial-non o :2 nw
(wx)b(wx)aXw sincos :2 wywy deceY
wywww ewxbwxayxu
w
)sin()cos(),(
0each for
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6.9 Dirichlet Problem for the Upper Half-Plane
x-xfxu
yx-yxu
for)()0,(
0 ,for0),(Consider the following problem: wy
www ewxbwxayxu
w
)sin()cos(),(
0each for
To satisfy the boundary condition we must generally superimpose these conditions over all w >= 0.
this is done by integration
0
),(),( dwyxuyxu w
0
)sin()cos( dwewxbwxa wyww
Boundary condition )()sin()cos()0,(
0xfdwwxbwxaxu ww
Fourier Integral expansion of f
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6.9 Dirichlet Problem for the Upper Half-Plane
x-xfxu
yx-yxu
for)()0,(
0 ,for0),(
Consider the following problem:
)()sin()cos(0
xfdwwxbwxa ww
Fourier Integral expansion of f
)()cos()(1
xfdwfaw
)()sin()(
1xfdwfbw
0
)sin()cos(),( dwewxbwxayxu wyww
0)( ))(cos(
1dwefdxw wy
dfdwexw wy )())(cos(1
0
0)sin())sin()(()cos())cos()((
1dwewxdwfwxdwf wy
0)( )sin()sin()cos()cos(
1dwefdwxwwxw wy
dfxy
y)(
)(
122
d
xy
fy22 )(
)(
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6.9 Dirichlet Problem for the Upper Half-Plane
y
x
x-xfxu
yx-yxu
for)()0,(
0 ,for0),(
Ex5/p279In each Problem obtain the solution of the Dirichlet problem for the upper half-plane, using the given f(x).
d
xy
fyyxu 22 )(
)(),(
cxck
cxxf
for
for0)(
0)( xf0)( xf
kxf )(
cc