Let C0 denote a positively oricntc ds .I ( '. ) - ;:;---:-

Chapter 0, Problem 12.1

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Let C0 denote a positively oricntc ds .I ( '. ) - ;:;---:- _]f I . C., .\' - :. expresses the value of f at any point :. interior to Co in terms of the values off al points son C0 In this section. we shall obtain from formula (I) a corresponding one for the real component of the function f: and. in Sec. 135. we shall use that resulc to solve the Dirichlet prohlcm (Sec. 116) for the disk bounded by Co. We let r0 denote the radius of C0 and \\Tite :. = r exp(iR ). where 0 < r < r0 (Fig. 191). The i11verse of the nonzero point:. with respect lo the circle is the point :. 1 lying on the same my from the origin as:.-. and satisfying the condition 1:.-. 1 11:.1 = re~. (Such inverse points. when r 0 = I. have already been used in Sec. 97.) Because 417 418 ll\TEGRAL foORMl.:L..\S Of' TllE POL~SOI'\ lYPE CllAP. 12 \' () FIGURE 192 (ro/r)>I. and this means that : 1 is exterior to the circle Co. According to the Cauchy-Goursat theorem (Sec. 50). then. I . /'(s) ds . = 0 . . c,, s - :1 Hence . r (:) = -. Ii (I --- I )1 . (.\')ls: I 2Jr I . ('11 .\' - :'. .\' - :'.I and. using the parametric representation s = r0 exp(i) (0 ~ < 2n) for C0 . we have ( 2) f (:) = _1 1 .!:r (-\ - - .\ ) f ( s ) d 2Jr o .\' - :'. S - :'. 1 where. for convenience. we retain the s to denote r0 cxp(i). Now and. in view of this expression for: 1 the quantity inside the paremheses in equation (2) can be wrillen - ) ) (3) .\' .\' : f(j - r- --- = -.--_ + :::;-----;: = . - ) . s - : .\' - .\' ( y /:.) .\ - " .\ - . 1.\ - . 1- An alternative fon11 of the Cauchy integral formula (I) is. thcretixe. ) ) ) .,,, f . ;11 rii - ,.-1-:r .f (roe'"") I . (re ) = ,., I . -12 l _Jr 0 .\ - . (4) when 0 < r < r0 . This form is also valid v . fo~n r = 0: in that case. it reduces directly to which is just the parametric fonn of equal.ion (I) when : = 0. SEC. 134 POISSOl\ li\TEGR.\L FORl\H'LA 419 The 4uantity Is - :: I is the distance bcl\\1ecn the points s and ::. and the law of cosines can be used to write (sec Fig. I 92) (5) ) ) ) Is - ::I-= r0 - 2ror cos( - 8) + ,.-. Hence. if 11 is the real component of the analytic function .f. it folhw.1s from l()rmula (4) that (6) I j . .:?,.,. (r(~ - r 2) u(ro, ) 11(r.8) = - , , d 2:r o r(j - 2ror cos( - (J) + r- ( r < ro). This is the Poisso11 integral fonnula for lhc hannonic func1ion 11 in the open disk bou ndcd by the ci re le r = ro. Formula (6) defines a linear integral transformation of u(ro. )into u(r. H). The kernel of the transformalion is. except for the factor I /(2;r ). the real-valued function ) ) (7) ,.- - ,.- P ( ro. r. - H) = 1 1 rii - 2rnr cos( - 8) + rwhich is known as the Poisso11 ker11el. In view of c4ua1ion (5 ). we can also \a.:ri le ) ) (8) ,.- - ,.- p ( r 0 . r. - H ) = 0 , . ls-::1- Lei us now verify the following properties of P. where r < ro: (a) Pis a positive function: ( ,\' +::) (h) P(r0 r. -8) =Re ~ : .\ .. (c) P(ro. r. - 8) is a hannonic func1ion of rand fJ imcrior to the circle Co for each fixed s on C0 : (d) P(ro. r. - 8) is an even periodic function of - H. with period 2:r: (e) P(r0 0. - H) = 1: I .:?..,. ( f) - f P(ro. r. - 8 )d = I when r < ro. 2][ Jo Property (a) is true because of expression (8) since r < r0 . Also. since :./(s - ::) and its complex conjugate :./CY - ;.) have the same real p - fJ) = Re ~ +- ~ =Re ~ . . \.. .\ .. .\ .. Thus P has properly (h): and since the real part of an analylic function is hannonic. P has property (c). As for properties (d) and (e). we find from expression (7) that P has those properties. Finally. property(.(} follows by writing u(r. H) =I in equation (6) and then referring 10 expression (7). 420 l!\TEGRAL fooRMt:L..\S or: TllE PoL~sol'\ lYPE CllAP. 12 \Ve conclude lhis imroduclion lo lhe Poisson imegral fonnula hy writing expression (6) as (9) 11 ( r. 8) = J lb 1 P ( r 0 . r.