On some three-harmonic potential for circular ring1. In this paper we shall give certain formulas for the 3-harmonic potential and for the circular ring

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXII (1981) ROCZNIKI POLSK1EGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXII (1981)

Ja n M u s i a e e k (Krakôw)

On some three-harmonic potential for circular ring

D = { X =

(xl f x2):

Let У = (yl t y2) and г2 ( X , У) = (x1- y 1)2+ (x2- y 2)2.

Let H ( Y ) = Л(f)L=r(o,r) be a continuous function for v e [ R l3 Rz]- The function

U 3(X; Y) = r4(X; -У) In г (X ; Y) is the fundamental solution for the equation A3u(X) = 0.

We shall examine the integral

w3{X) = f f H ( r ) U 3(X; Y)dY

D

of the potential type.

2. Let t = £ (OX; 0У) and v = r(0, У ) е [ Я 1} R2] be polar coordinates of the point У.

We have

(1) r2(X ; У ).= г2(х 150; У) = v2 + x 2 — 2vxx cos t.

By Budak et all. О and from (1) we obtain

00

(2) — ln r (X ; У) = — In x j T £ n~x ((x j)-1 vf cos nt for x 2 > R 2 n — 1

and

00

(3) — In г (X ; У) = -ln t> + £ n~1 ((f)-1 Xi)" cos nt for < R ^

n= 1

If x l e { R 1, R 2), then for v e ( R 1, x 1) holds formula (2), and for г е (х 1, Я 2) holds formula (3). (*)

(*) В. M. B u dak, A. A. S a m a r sk i and A. N. T ic h o n o w , Z adania i pro ble m y f i z y k i mat em at yc zn ej , P W N , Warszawa 1965, p. 475.

306 J. M u sia le k

Remark. For every point P for which |0P| = x x we have w3 (P)

= vv3(x1,0).

Consequently we can suppose that P = (x^O).

Now we shall prove three theorems.

Th e o r e m 1. Let x x > R 2 and the function h(v) be continuous for rG [R l5 Rf\; then

R2

w3(xt) = — 2л(2 + 1пх1) J h(v)v5dv — 2кх\ (1 +4 In X j)x

r2 r

2

x j h(v)v3 dv — 2Kxf In x t { h(v)vdv.

Ri «1

P roof. Use in the integral w3 polar coordinates v, t and by formula (2) our theorem is proved.

Th e o r e m 2. I f the function h(v) is continuous for u e [R 1 }.R2] and e(0, R j), then

«2 ' « 2

w3(xi) = — 2тг J h(v)v5 In vdv — 2nx^ J h(v)v\nvdv —

Rl *i

r2 R2 r 2

— 4 nx\ J h (v) v3 dv — Snxj J h(v) v3 In vdv — 2nx* J h(v)vdv.

R l R i «1

P roof. Applying in the integral w3 the polar coordinates v, t by formula (3) we obtain the thesis of Theorem 2.

From Theorems 1 and 2 we can easily deduce the following

Th e o r e m 3. I f the function h(v) is continuous for r e [ R l5 R2] and Xi e ( R l , R 2), then

w3(x i) = — 2rc(2 + ln x 1) J h(v)v5 dv — 2кх\ (1 + 4 In x t) x

*i

X1 X1 R2

x J h(v)v3 dv — 2nx\ In x 1 J h(v)vdv — 2n J h(v)v5 In vdv —

Ri Ri xi

r 2 r 2

— 2nxi J h(v)v\nvdv — $nxj J h(v)v3 \nvdv —

xi xi

r 2 r 2

— 4nxj J h(v)v3dv — 2nx‘l J h(v)vdv.

*i xi