By H. Begehr
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Extra resources for Complex Analytic Methods for Partial Diff. Eqns. - An Intro. Text
Let G E C(r; if) and G(C) # 0 on r. Then the index K of G with respect to r is the mean variation of arg G(() while ( varies on r in the positive direction passing any point once, tc := indG = 2s J d argG(() = 2' r Jr dlog G(C) . Because r is closed and G is continuous K is an entire number. The index has the following properties. 1. ind (G1G2) = ind G1 + ind Gs, ind(1/G) = -ind G = ind G . 2. If D is a domain with smooth boundary and G is an analytic function in D up to isolated poles with continuous non-vanishing boundary values on 8D then indG = n(0) - n(oo) , where n(0) is the number of zeroes and n(oo) the number of poles of G each counted with respect to its multiplicity.
BERGMAN, see [Berg50,Besc53,Cour50J. Because g(z, () vanishes for C E aD identically in z E D a9(z,C) Oz = 029(z,CC,(s) °, aza( + a29(z,C('(s) ata( = 0, C = C(s) E aD , Function theoretical tools 35 so that for z E D, ( E 8D L(z, ()('(s) + K(z,C)t'(s) = 0 . Hence, Ju()L(z,c)d(. u=(z) = 2i 8D As u is harmonic u= is analytic. Realizing that u is real so that us = uz this leads to Ju(c)dc + -C(C)d-} = 2 f Re uc(t; )dr; u(z) - u(a) = a a u(() / L(t, ()did( to aD a Introducing Jt(z,() f L(t,()dt , JK(z,() JK(t,)dt a a this becomes u(z) = Re f Jt(z, ()u(()d( + u(a) .
Let now w(zo) = wo for some zo E D. Then w(z) -wo = Iwol exp(i(argwo + pr)) . This holds because w is one of the branches of the square root. Hence, there exists a c E (V satisfying w(z) # c in D. Moreover, because we may assume c E -w[D] and because -w[D] is an open set there exists an rr > 0 such that 0 < n < I w(z) - cI in D. 't'hen we may choose A, B such that w1(z1) = 0, Iwi(z)I < I for some fixed z, E D and all z E D. It is easy to see that w1ES. Complex Analytic Methods for Partial Differential Equations 20 2.