Let f be an analytic funclion of:: lhroughout the half

Chapter 0, Problem 12.17

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Let f be an analytic funclion of:: lhroughout the half plane Im:: > 0 such thal for some positive conslants ll and M. lhe order property ( I ) (Im::~ 0) is satisfied. For a fixed point:: above the real axis. let CR denole lhe upper half of a positively orierllec.J circle of mdius R cenlered m the origin. where R > 1::1 (Fig. I 98). Then. according to the Cauchy integral fonnula (Sec. 54) . . __ -'-1 f(s) ds -'-JR fU) dt ( 2) .f ( ,, ) - ") . . . - + ") . - . _'Jf I < K .\ - ,, _'Jf I . R I - . SEC. 138 SCllWARZ )l\TEGR.\L FORMlL\ 429 \' x FlGURE 198 We find chm che lirsl of chcse imcgrnls approaches 0 as R lends co oo since. in vicv .. of condicion (I), Thus I ~ f .. (\_') ,_1s I < __ ,\!,_1 __ JT R = __ JT_1\!,_f -- (;: ' . R'j(R - 1:.1> R'J( I -1:.1/ R) ___ I_ ;:x- fU) dt /L) - -, . _JT I "'- t - :. (Im:. > 0). Condicion (I) also ensures char chc improper incegrnl here converges.~ The number co which il converges is chc same as ics Cauchy principal value (sec Sec. 85). and rcprcscmacion (3) is a Ct111chy i11t(~rttl.fim1111/afor the lw~f plane Im:. > 0. When chc point :. lies below che real axis. chc righc-hand side of cquacion (2) is zero: hence inccgml (3) is zero for such a poinc. Thus, when:.. is above chc real axis. we have che following expression. where c is an arbicrnry complex conscanc: (4) I /'" ( I c ) f (:.) = :;-:- ---=-=- + _ -=- f (I) d t (Im:. > 0). _JT I . . . "'- t . t .. In che two cases c = - I and c = I. chis reduces. rcspcccivcly. co (5) -- ~ J '.'.\:,; yf (I ) /(,,) - ) dt JT "'- It - :.I- ( y > 0) and (6) . - - _I /" U - .r)/(f) .f ( ,, ) - . I -1) d t JT I .. "'- t - . - (y > 0). If f(:.) u(x. y) + iv(x. y). il follm\1s from cquacions (5) and (6) thac chc harmonic funccions 11 and t~ arc rcprcscmcd in chc half plane y > 0 in ccrms of chc boundary values of 11 by chc expressions (7) I/" y11(1.0) _ ~ ;"'- y11(1.0) II (X . ,\') = - , d t - , 1 d t n "'- lt-:.1- n. "'-(1-x)-+y- (y > 0) 'Sec. for installl.'C. A. E. Taylor an 0). Expression (7) is known as theSclnvarzi11tegralformula. or che Poisson imegral formula for the half plane. In chc next section. we shall relax chc conditions for chc vali