Contest Problem Book IV: Annual High School Examinations, by Ralph A. Artino

By Ralph A. Artino

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Extra resources for Contest Problem Book IV: Annual High School Examinations, 1973-1982

Sample text

Presque partout). H. e. as r -^ oo. COROLLARY. 23 ) with the same value of ca as u. Given any positive ε, we write Φ(Γ) = cos (πλ) B(r, u) — A(r, u), Ε(ε) = {r|(r) > srx and r > 1}. 8, Γ JE(e) dr r

Suppose then that Φ(Γ0) = M. 16). 9). 6, and so U(z) is extremal under these hypotheses. It is not difficult to see that the (7(rei0) for fixed Θ are the only extremal functions, and in particular that U(z) is uniquely defined by (a)-(d). Suppose in fact that u(z) is extremal for some r = r0. e. B(r0,u)

In the plane. Thus if z = x + \y = iRZ then we have v(z) = υ(-Κη)Κ1(Ζ9η)άη + v(Relt)K2(Z, t - \n)àt. 23). 23), we see that i;(Rnew) = 0 ( R i ) as n -► oo uniformly in |0| ^ π. H. 3 as R = Rn -> oo. 21) i i i;(-Äif)K1(Z,V)dij = ^ j rR /(ΟκΥ-^,-^Ιαί. 22) = l(\z -t\~2- 1 Κ,ί^,^Λ R2\tz - R2r2). 23) with finite a, i: |»(f)l + |p(-t)l}