Mathematical Physics Vol 1

7.7 Green formulas

377

Substituting these values into (7.269) we obtain

+ π Z

+ π Z

+ ∞ ∑ n = 1

r n R n

1 2 π

1 π

u ( r , ϕ )=

f ( t ) cos n ( t − ϕ ) d t ,

f ( t ) d t +

(7.272)

− π

− π

or

" 1 + 2

n cos n ( t − ϕ ) # f ( t ) d t .

+ π Z

+ ∞ ∑ n = 1

r R

1 2 π

u ( r , ϕ )=

(7.273)

− π

However, as

∞ ∑ n = 1

1 − c 2 c 2 − 2 c cos ϕ + 1 ,

c n cos ( n ϕ )=

1 + 2

0 ≤ c ≤ 1 ,

(7.274)

we finally obtain the solution of the Dirichlet problem for a circle

+ π Z

R 2 − r 2 R 2 − 2 rR cos ( t − ϕ )+ r 2

1 2 π

u ( r , ϕ )=

f ( t )

d t .

(7.275)

− π

This relation is known in literature also as the Poisson integral . Dirichlet problem for a sphere

In this case, the problem is reduced to finding a function u that satisfies the Laplace equation ∆ u = 0, and whose value on the sphere S is known ( u | S = f ). Observe now the sphere S , with radius R , and two points: A ( a , b , c ) in the interior of the sphere and A 1 ( a 1 , b 1 , c 1 ) out of the sphere. Let O be the center of the sphere, and point M ( x , y , z ) ∈ S (Fig. 5.4).

Figure 7.4:

Let us introduce the following notation

AM = r , A 1 M = r 1 , OA = l , OA 1 = l 1

(7.276)

A

M

A

r

r

1

l

r

α

r

A

M

O

M

A

O

l

R

1

1

1

1

Figure 7.5:

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