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Download PDF by W. E. Kirwan, L. Zalcman: Advances in complex function theory: Proceedings of seminars

By W. E. Kirwan, L. Zalcman

ISBN-10: 3540075488

ISBN-13: 9783540075486

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It remains finite. to show that It is clear domain D where the total number that each v < 0. F Further, Since the total n u m b e r of tracts of p of curves separates D all these D9 -v F is from at least one are distinct. is finite, p must be finite. L EMM A ber. 2. 2) z onto : w0 v D be any c o m p l e x nump roots in D, as imaginary part. by a c o n f o r m a l mapping t(~), and set F(~) Then U is p o s i t i v e = f{t(~)} and h a r m o n i c in = U + iV. 2). 13, that p. 179) f r o m the imaginary F(~) = ic ~=i where the Thus c are r e a l F(~) evidently tion f(z) Lemma 2.

T 2 = x ~+I/R2 Rkp~l)(t ) is clearly Similarly a homogeneous if k Rk_(~ r k )(t) where ~ homogeneous is a polynomial polynomial It remains x = Xm+ I, : = is a polynomial of Thus 89 cv x2vm+IRk-2v ~=0 of degree k in xI to Xm+ I. k = 2p + i, RtR2P~(t 2) of degree of degree to show that p2 = [mv:l xg, 2 P~I) by Lemma 4. polynomial is odd, Then v k = Xm+iR2Pr p, so again in xI is harmonic. t : x/R, Q = Rkp(x/R) to Rkp~l)(t) is a Xm+ I. To see this, we write 36 where P = P~k). 7) ~-~ P(t) and for 9 : !

Ero8 equation 35 pkk)(t)c Also k is an even or odd f u n c t i o n of t c o r r e s p o n d i n g as is even or odd. The polynomials nomials. 4), ultraspherical is discussed poly- on p. 117 of [14] on p. 5). The last p. 80. We deduce LEMMA 5. u = U(Xl,X2,''',x m) Let nomial of degree xI in m+l x~) 89 R = ([v=l and set v v ~ = be a homogeneous x m. to Let t = Xm+i/R. be a positive integer Then if ~k~(1) ~ ~k (t)u(x), is a homogeneous k harmonic poly- 1 1 ~ + ~ (m-l] = harmonic p o l y n o m i a l of degree s + k in xI to Xm+ 1 9 Suppose first that ~i k degree in k is even.

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Advances in complex function theory: Proceedings of seminars held at Maryland University 1973 74 by W. E. Kirwan, L. Zalcman

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